Short answer
A conscience tells us that right and wrong exist, but it does not reliably tell us what they are. People fill in the details from their own time and interests, and they do it with a clear conscience. A young man raised in Nazi Germany and a farmer raised in America in the 1940s were both sure they knew good from evil. They did not agree, and the first was terribly wrong. The medieval Church called lending at interest a grave sin. Today the same countries run on it. The Qur'an forbade it in the seventh century, and it forbids it still. Islam teaches that God (﷾) gave us the moral sense. Through revelation He also gives the standard it answers to. That standard is the same for rich and poor, and for every century.
Full explanation
The moral sense is built in
The Qur'an says God (﷾) showed every soul its wrong and its right when He formed it. It calls this inborn nature the fitra (the pattern on which God (﷾) made mankind). The Prophet Muhammad (ﷺ) said every child is born upon the fitra, and that parents then shape the child's religion. So the sense that right and wrong exist is given, not learned.
“By the Soul, and the proportion and order given to it; And its enlightenment as to its wrong and its right;-”
“So set thou thy face steadily and truly to the Faith: (establish) Allah's handiwork according to the pattern on which He has made mankind: no change (let there be) in the work (wrought) by Allah: that is the standard Religion: but most among mankind understand not.”
Abu Hurayra reports that the Prophet (ﷺ) said every child is born upon the fitra (the natural disposition God gave mankind), and it is his parents who make him a Jew, a Christian or a Magian, just as an animal is born whole. Abu Hurayra then recited Qur'an 30:30.
The Bible agrees that the law is written in the heart
Paul, a Christian missionary of the first century, says that people without the law still show its work written in their hearts. Their conscience bears witness to it. Christians read this as the conscience God (﷾) gave to all people. Muslims agree that the sense is universal.
“For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves: Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another;”
The Bible's own warning about a man's own sense of right
The Hebrew Bible warns that a way can seem right to a man and still end in death. Judges sums up an age of disorder with the words that every man did what was right in his own eyes. A man's own sense of right is not a safe guide.
“There is a way which seemeth right unto a man, but the end thereof are the ways of death.”
“In those days there was no king in Israel: every man did that which was right in his own eyes.”
A clear conscience that changed its mind
Through most of the Middle Ages the Church condemned lending money at interest as a grave sin. In 1311-1312 the Council of Vienne (a gathering of Catholic bishops) made a harsh ruling. Anyone who stubbornly held that interest was no sin was to be punished as a heretic (someone who holds a belief the Church has condemned). Today the same countries run on interest, and hardly a pulpit in them calls it a sin. Sincere people held both views. Conscience did not settle which was right. The Church drew its ban from the Bible itself, and the ban still went. A standard helps only if it is kept as it was given.
The council decrees that anyone who obstinately maintains that practising usury is not a sin is to be punished as a heretic.
“If thou lend money to any of my people that is poor by thee, thou shalt not be to him as an usurer, neither shalt thou lay upon him usury.”
Traces Catholic teaching on usury from the medieval condemnation of lending at interest to its gradual relaxation in later centuries, when moderate interest came to be allowed.
A standard that has not moved
The Qur'an permits trade and forbids riba (usury, interest on a loan). It said so in the seventh century. Its words on riba are unchanged, and Muslim scholars of every school still read them as a ban. A law that comes from God (﷾) does not bend to whoever holds power, or to the fashion of an age.
“Those who devour usury will not stand except as stand one whom the Evil one by his touch Hath driven to madness. That is because they say: "Trade is like usury," but Allah hath permitted trade and forbidden usury. Those who after receiving direction from their Lord, desist, shall be pardoned for the past; their case is for Allah (to judge); but those who repeat (The offence) are companions of the Fire: They will abide therein (for ever).”
Perspectives
Mainstream Christian
God wrote His law on the human heart, and the conscience bears witness to it. Scripture is still needed to inform and correct the conscience.
Paul teaches that people without the written law still have its work written in their hearts. Their conscience bears witness to it.
Rebuttal
Islam agrees that God (﷾) put this sense in every soul (91:7-8), and that revelation must correct it. The question is which revelation's standard has been kept as it was given. The Church's ban on interest went, but the Qur'an's ban on riba has not moved.
Scripture warns that a way can seem right to a person and still lead to death.
Rebuttal
Islam agrees that a person's own sense of right is not a safe guide. Islam goes further. The Qur'an gives a standard, such as its ban on riba, that has not moved with the times.
Secular
Conscience and reason are enough; people can work out right and wrong together without scripture.
People can be good without religion, guided by conscience and reason.
Rebuttal
Islam agrees that people do much good by conscience and reason, because God (﷾) placed the fitra, a natural sense of right, in every soul. But sincere, reasoning people reach opposite answers. And if right is only what an age agrees on, no age can be wrong. A standard from God (﷾), kept as it was given, does not move with any age. The Qur'an's ban on riba shows this.
The Islamic perspective
God (﷾) placed the fitra in every soul, so all people know that right and wrong exist. Revelation gives the fixed standard that the fitra answers to. It is the same for every people and every century.
The Qur'an teaches that God (﷾) formed every soul. When He formed it, He gave it an awareness of its wrong and its right.
The Prophet (ﷺ) said every child is born upon the fitra. His parents then make him a Jew, a Christian or a Magian (a Zoroastrian, a follower of the old Persian religion).
Sincere consciences have reversed themselves on a moral question. In 1311-1312 the Council of Vienne ruled that anyone who stubbornly denied that usury was a sin should be punished as a heretic. Christian countries later came to accept interest.
The Qur'an forbade riba, or interest, in the seventh century, and its text forbids it still. So its standard has not moved with the times.
If morality is whatever each generation agrees on, no generation can be wrong. So a fixed standard must come from outside human opinion.