Short answer
Islam gives its people one name: Muslim. The Qur'an says God (﷾) Himself named them Muslims. It tells believers to hold on to Allah's (﷾) rope all together and not to split apart. It tells the Prophet (ﷺ) that he has no part in those who break their religion into sects. Even the honoured names of the Prophet's (ﷺ) companions, the Ansar of Medina and the Emigrants from Mecca, were not allowed to become rival cries. When two men shouted them as party calls in a quarrel, the Prophet (ﷺ) stopped them. So a Muslim needs no other label. Muslim is the whole title.
Full explanation
A name given by God (﷾)
The Qur'an says God (﷾) Himself named the believers Muslims, before the Qur'an and in it. The verse says "He". Most commentators, Ibn Abbas among them, read this as God. Ibn Abbas was a cousin and companion of the Prophet (ﷺ) in the seventh century. Either way, the name comes from revelation.
“And strive in His cause as ye ought to strive, (with sincerity and under discipline). He has chosen you, and has imposed no difficulties on you in religion; it is the cult of your father Abraham. It is He Who has named you Muslims, both before and in this (Revelation); that the Messenger may be a witness for you, and ye be witnesses for mankind! So establish regular Prayer, give regular Charity, and hold fast to Allah! He is your Protector - the Best to protect and the Best to help!”
Proud of the one name
The Qur'an counts among the best speakers the one who calls people to God (﷾) and does good. That person says plainly that he is one of those who bow in Islam, a Muslim. No other label is added to it.
“Who is better in speech than one who calls (men) to Allah, works righteousness, and says, "I am of those who bow in Islam"?”
The command not to divide
The Qur'an commands believers to hold fast to Allah's (﷾) rope all together and not be divided. The rope is a picture of staying joined to His guidance. The Qur'an reminds them that God (﷾) joined their hearts after they had been enemies. It tells the Prophet (ﷺ) that he has no part at all in those who split their religion into sects, which are rival groups. It describes such people as parties, each pleased with what it has.
“And hold fast, all together, by the rope which Allah (stretches out for you), and be not divided among yourselves; and remember with gratitude Allah's favour on you; for ye were enemies and He joined your hearts in love, so that by His Grace, ye became brethren; and ye were on the brink of the pit of Fire, and He saved you from it. Thus doth Allah make His Signs clear to you: That ye may be guided.”
“As for those who divide their religion and break up into sects, thou hast no part in them in the least: their affair is with Allah: He will in the end tell them the truth of all that they did.”
“Turn ye back in repentance to Him, and fear Him: establish regular prayers, and be not ye among those who join gods with Allah,- Those who split up their Religion, and become (mere) Sects,- each party rejoicing in that which is with itself!”
Even honoured names may not become banners
The Ansar were the Muslims of Medina who took the Prophet (ﷺ) in. The Emigrants were the Muslims who left Mecca with him. The Qur'an praises both groups by name (in Arabic, al-muhajirin and al-ansar). Yet on one campaign a quarrel grew. One man shouted for the Ansar, and another shouted for the Emigrants. The Prophet (ﷺ) called it the call of the age of ignorance, the time before Islam, and told them to stop. If the names of his own companions could not be party cries, no later name can be.
“The vanguard (of Islam)- the first of those who forsook (their homes) and of those who gave them aid, and (also) those who follow them in (all) good deeds,- well-pleased is Allah with them, as are they with Him: for them hath He prepared gardens under which rivers flow, to dwell therein for ever: that is the supreme felicity.”
Jabir reports that on a campaign an Emigrant struck a man of the Ansar in jest. The Ansari cried out for help to the Ansar, and the Emigrant cried out for help to the Emigrants. The Prophet (ﷺ) came out and asked why they were raising the call of the age of ignorance, and told them to stop, for it was a foul call.
Perspectives
The Islamic perspective
Believers carry the one name God gave them, Muslim. The Qur'an and the Prophet (ﷺ) forbid them to split into rival camps under other names. No label, however honourable, may become a banner that divides them.
The Qur'an states that God (﷾) named the believers Muslims, both before the Qur'an and in it.
The Qur'an commands believers to hold fast to Allah's (﷾) rope together and not be divided.
The Qur'an tells the Prophet (ﷺ) that he has no part in those who split their religion into sects.
The Qur'an praises the one who calls to God (﷾) and does good. He says he is one of those who bow in Islam, the Muslims.
The Prophet (ﷺ) stopped his companions using the names Ansar and Emigrants as rival cries. He called it the call of the age of ignorance.