Short answer
No. Isaiah 53 never names the servant. Isaiah uses the word "servant" for more than one figure, and Jewish readers have long read this chapter as speaking of Israel. When Matthew quotes its words, he uses them for Jesus healing the sick. The Bible also says that every man shall be put to death for his own sin. The son shall not bear the iniquity of the father. Jesus quoted God's (﷾) words, "I will have mercy, and not sacrifice." In the Gospels, the priests plotted to kill him, and Pilate found no fault in him. By the Gospels' own account, it was the killing of an innocent man. The Qur'an says the plotters failed. They did not kill him, and they did not crucify him.
Full explanation
The verse and the Christian reading
Christians read Isaiah 52:13 to 53:12 as a prophecy of the Messiah, God's (﷾) promised king. They say he would suffer and die in the place of sinners. They point to Acts, where Philip, an early Christian preacher, starts from this passage and preaches Jesus to an Ethiopian official.
“But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed.”
“Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.”
The Jewish reading: the servant is Israel
In several of the chapters that lead up to Isaiah 53, God (﷾) calls Israel His servant by name. In the Jewish reading, the speakers in chapter 53 are the kings of the nations. They are astonished to find that the people they despised was God's (﷾) servant. The JPS 1917 translation (the Jewish Publication Society's English Bible) renders the Hebrew as wounded "because of" our transgressions. In the Jewish reading the nations confess that Israel suffered because of their sins, at their hands. The chapter names no one, and says nothing of God (﷾) or a Messiah dying.
“But thou, Israel, My servant, Jacob whom I have chosen, The seed of Abraham My friend;”
“Yet now hear, O Jacob My servant, And Israel, whom I have chosen;”
“And He said unto me: ‘Thou art My servant, Israel, in whom I will be glorified.’”
“So shall he startle many nations, Kings shall shut their mouths because of him; From that which had not been told them shall they see, And that which they had not heard shall they perceive.”
“But he was wounded because of our transgressions, He was crushed because of our iniquities: The chastisement of our welfare was upon him, And with his stripes we were healed.”
An early record of the Jewish reading
The Jewish reading is not a modern answer to Christians. Origen, a Christian scholar writing around 248 CE, records a debate with a learned Jew. The Jew took the passage as speaking of the whole people of Israel. They were suffering and scattered among the nations.
Origen, a Christian scholar, records that when he quoted Isaiah 53 in a debate with learned Jews, his Jewish opponent answered that the prophecy speaks of the whole Jewish people as one person, scattered among the nations and suffering.
How Matthew uses Isaiah 53
Matthew quotes Isaiah 53:4 when Jesus heals the sick and casts out spirits one evening. For Matthew, the servant "bare our sicknesses" by taking them away in healing, during his ministry, not by dying.
“When the even was come, they brought unto him many that were possessed with devils: and he cast out the spirits with his word, and healed all that were sick: That it might be fulfilled which was spoken by Esaias the prophet, saying, Himself took our infirmities, and bare our sicknesses.”
Every soul carries its own burden
The Law of Moses and the prophet Ezekiel both say that no one is put to death for another's sin. The Qur'an says the same: no one can carry another's burden, not even a close relative. Moses once offered to be blotted out in Israel's place. God (﷾) refused and gave a rule: whoever has sinned against Him is the one blotted out. The Law also forgave a poor man's sin with an offering of flour, with no blood at all. It was a tenth of an ephah, a dry measure.
“The fathers shall not be put to death for the children, neither shall the children be put to death for the fathers: every man shall be put to death for his own sin.”
“The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.”
“Nor can a bearer of burdens bear another's burdens if one heavily laden should call another to (bear) his load. Not the least portion of it can be carried (by the other). Even though he be nearly related. Thou canst but admonish such as fear their Lord unseen and establish regular Prayer. And whoever purifies himself does so for the benefit of his own soul; and the destination (of all) is to Allah.”
“Yet now, if thou wilt forgive their sin—; and if not, blot me, I pray thee, out of thy book which thou hast written. And the Lord said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.”
“But if he be not able to bring two turtledoves, or two young pigeons, then he that sinned shall bring for his offering the tenth part of an ephah of fine flour for a sin offering; he shall put no oil upon it, neither shall he put any frankincense thereon: for it is a sin offering. Then shall he bring it to the priest, and the priest shall take his handful of it, even a memorial thereof, and burn it on the altar, according to the offerings made by fire unto the Lord: it is a sin offering. And the priest shall make an atonement for him as touching his sin that he hath sinned in one of these, and it shall be forgiven him: and the remnant shall be the priest’s, as a meat offering.”
Mercy, not sacrifice
Jesus told the Pharisees, the strict teachers of the law, to go and learn what God (﷾) meant by "I will have mercy, and not sacrifice". He was quoting the prophet Hosea. God (﷾) asks for mercy and for knowledge of Him rather than offerings. In Jesus' own parables the prodigal son is forgiven the moment he returns. The tax collector who only asks for mercy goes home justified. Neither story has a sacrifice. The page on Hebrews 9:22 gives the full argument.
“But go ye and learn what that meaneth, I will have mercy, and not sacrifice: for I am not come to call the righteous, but sinners to repentance.”
“For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.”
“And he arose, and came to his father. But when he was yet a great way off, his father saw him, and had compassion, and ran, and fell on his neck, and kissed him.”
“And the publican, standing afar off, would not lift up so much as his eyes unto heaven, but smote upon his breast, saying, God be merciful to me a sinner. I tell you, this man went down to his house justified rather than the other: for every one that exalteth himself shall be abased; and he that humbleth himself shall be exalted.”
What the Gospels say happened
In the Gospels, the people who brought about his death were plotting a murder. The chief priests met at the high priest's palace. They planned to seize Jesus by trickery and kill him. Pilate said more than once that he found no fault in him, and gave way only to stop a riot. The night before, Jesus had prayed for the cup to pass from him. Nothing in the Law of Moses makes the killing of an innocent man by his enemies an offering to God (﷾). John has Jesus say he lays his life down of himself. But a willing victim does not make it just to punish the innocent for the guilty.
“Then assembled together the chief priests, and the scribes, and the elders of the people, unto the palace of the high priest, who was called Caiaphas, And consulted that they might take Jesus by subtilty, and kill him.”
“Then said Pilate to the chief priests and to the people, I find no fault in this man.”
“Pilate saith unto him, What is truth? And when he had said this, he went out again unto the Jews, and saith unto them, I find in him no fault at all.”
“When Pilate saw that he could prevail nothing, but that rather a tumult was made, he took water, and washed his hands before the multitude, saying, I am innocent of the blood of this just person: see ye to it.”
“And he went a little further, and fell on his face, and prayed, saying, O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt.”
“No man taketh it from me, but I lay it down of myself. I have power to lay it down, and I have power to take it again. This commandment have I received of my Father.”
The plotters failed
The Qur'an states that those who boasted of killing Jesus did not kill him or crucify him. It was only made to appear so to them.
“That they said (in boast), "We killed Christ Jesus the son of Mary, the Messenger of Allah";- but they killed him not, nor crucified him, but so it was made to appear to them, and those who differ therein are full of doubts, with no (certain) knowledge, but only conjecture to follow, for of a surety they killed him not:-”
Verse by verse
Isaiah 53:5 — Nevi'im, Yeshayahu, chapter 53, verse 5
“But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed.”
Cited for
The servant was punished in our place. His wounds paid for our sins, and by them we are healed.
The response
Isaiah 52:15 introduces the nations and their kings as the ones who are astonished. In the Jewish reading, they are the speakers of chapter 53. That reading, drawn from Isaiah's own words in earlier chapters, takes the servant here to be Israel. In that reading, "my people" in verse 8 means the nations' own peoples, since they are the ones speaking. The Hebrew can be read, as the JPS 1917 renders it, "because of our transgressions": the servant suffered at the hands of sinners. And the Law forbids Israel's judges to put anyone to death for another's sin. Through Ezekiel, God (﷾) gives the same rule for His own judgment: "The soul that sinneth, it shall die."
Where it lands — The verse describes a righteous sufferer, and it names no one. Read with the rest of Isaiah and the Law, it does not teach that God (﷾) punishes the innocent for the guilty.
Isaiah 53:10 — Nevi'im, Yeshayahu, chapter 53, verse 10
“Yet it pleased the Lord to bruise him; he hath put him to grief: when thou shalt make his soul an offering for sin, he shall see his seed, he shall prolong his days, and the pleasure of the Lord shall prosper in his hand.”
Cited for
God made the servant's soul "an offering for sin", so the chapter describes an atoning sacrifice.
The response
The JPS 1917 translation reads the Hebrew as a condition: "to see if his soul would offer itself in restitution". The same verse promises the servant "seed" and a prolonged life after his suffering. In the Jewish reading this is a people who outlive persecution and have descendants. And the Law forbids Israel's judges to put anyone to death for another's sin. Through Ezekiel, God (﷾) gives the same rule for His own judgment: the son shall not bear the iniquity of the father.
Where it lands — The verse does not name Jesus. Read with Ezekiel, it cannot mean that God (﷾) accepts the death of an innocent man for the guilty.
Isaiah 53:11-12 — Nevi'im, Yeshayahu, chapter 53, verses 11-12
“He shall see of the travail of his soul, and shall be satisfied: by his knowledge shall my righteous servant justify many; for he shall bear their iniquities. Therefore will I divide him a portion with the great, and he shall divide the spoil with the strong; because he hath poured out his soul unto death: and he was numbered with the transgressors; and he bare the sin of many, and made intercession for the transgressors.”
Cited for
Verses 11 and 12 say the servant bore the sin of many, which is substitution.
The response
When Moses offered himself in Israel's place, God (﷾) refused: "Whosoever hath sinned against me, him will I blot out of my book" (Exodus 32:33). Through Ezekiel God (﷾) states His rule for His own judgment: "The soul that sinneth, it shall die" (Ezekiel 18:20). Whatever the servant bore, it was suffering at sinners' hands. It was not God punishing an innocent man in the guilty's place.
Where it lands — The chapter says the servant suffered because of others' sins. When Moses offered himself in Israel's place, God (﷾) refused and gave His rule: whoever has sinned is the one blotted out.
Acts 8:35 — Acts of the Apostles, chapter 8, verse 35
“Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.”
Cited for
The New Testament itself reads Isaiah 53 as the death of Jesus for sins.
The response
Acts tells us what Philip preached, not what Isaiah meant. The chapter never names Jesus. When Matthew quotes Isaiah 53:4, he uses it for healing the sick. When Jesus quotes the chapter in Luke 22:37, his words are "he was reckoned among the transgressors". That means being counted with criminals, not paying for sins. Letters such as 1 Peter do read the chapter as his death for sins, but that is their reading, not words Isaiah wrote. Even readers who see a single person here must add the name Jesus, because Isaiah never supplies it.
Where it lands — Where the Gospels quote Isaiah 53, none of them uses it to say God (﷾) punished Jesus for others' sins. Matthew uses it for healing, and Luke for being counted among criminals.
Perspectives
Mainstream Christian
Christians say Isaiah 52:13 to 53:12 foretells the Messiah, who suffers and dies for the sins of others. They say the New Testament applies it to Jesus, and that his death is the sacrifice by which sinners are healed.
Isaiah 53:5 says the servant was wounded for our transgressions. Christians read this as Jesus dying in the place of sinners.
Rebuttal
The verse names no one. The Hebrew can be read, as the JPS 1917 renders it, "because of our transgressions". And the Law and Ezekiel say no one is put to death for another's sin.
In Acts, Philip begins from Isaiah 53 and preaches Jesus.
Rebuttal
Acts tells what Philip preached, not what Isaiah meant. Matthew uses Isaiah 53:4 for healing the sick, and Luke has Jesus use the chapter for being counted among criminals.
Jewish
The servant is Israel, whom Isaiah calls "my servant" by name in the chapters before. In chapter 53 the nations confess that the people they despised and oppressed was God's servant all along.
Isaiah names the servant as Israel more than once in the chapters before 53.
Rebuttal
Islam agrees that Isaiah calls Israel "my servant" in several places, and that chapter 53 never names Jesus. But Isaiah uses "servant" for more than one figure, as chapter 42 shows. Islam goes further: even if one person is meant, the Law and Ezekiel say no one carries another's sin.
The JPS 1917 translation has the servant wounded "because of" the speakers' transgressions. In the Jewish reading the nations confess that Israel suffered because of their sins, at their hands. The chapter names no one, and says nothing of God or a Messiah dying.
Rebuttal
Islam agrees that the verse can be read "because of our transgressions": the servant suffered at sinners' hands, not in their place. Islam goes further. The Qur'an says no soul bears another's burden, and that Jesus was neither killed nor crucified.
A learned Jew debated Origen in the third century. Already then he read the passage as speaking of the whole people of Israel.
Rebuttal
Islam agrees that this shows the Jewish reading is old, and not a modern reply to Christians. Islam goes further: whoever the servant is, the Law and Ezekiel say no one is put to death for another's sin.
The Islamic perspective
Isaiah 53 never names Jesus. Read with the Law, it does not teach that God (﷾) punishes an innocent man for the sins of others. Moses, Ezekiel and the Qur'an all say each soul carries its own burden. By the Gospels' own account the death was a plot and an unjust verdict, and the Qur'an says the plot failed.
The Law and the prophets say that no one is put to death for another's sin.
When Moses offered himself in Israel's place, God (﷾) refused a substitute. The Law forgave a poor man's sin with an offering of flour and no blood.
Jesus quoted Hosea: God (﷾) wants mercy, not sacrifice.
Matthew applies Isaiah 53:4 to Jesus healing the sick, not to a death for sins.
In the Gospels the chief priests plotted to kill Jesus. Pilate found no fault in him, and gave way only to avoid a riot.
Punishing an innocent man for the guilty is injustice, whether or not he is willing.
The Qur'an teaches that no one bears another's burden, and that Jesus was neither killed nor crucified.