The mystery of Jesus' identity begins with his very name. "Jesus Christ", or a Paul calls him, "Christ Jesus" was certainly not the real name of the founder of Christianity...
See, this is the sort of nit picky stuff that hurts the credibility of those trying to debunk the Bible.
Yes and no. Firstly, it is possible that Jesus/Joshua (etc.) had a different birth name. It is possible that he chose or was given the name Jesus/Joshua in later life. It is also possible that the name Jesus/Joshua was attributed to him after his death. I have no view on this as I do not see it is as fundamental to whether or not he existed.
For comparison, there was no one called Confucius. That name is a Latinised version of "Kǒng Fūzǐ" which, in turn, means "Master Kong" — the "Master" part of which is clearly an honorific. This does not mean that Confucius does not exist — only that we know him by a different name from the one he used in life.
Indeed, the idea of a person having one name, linked in some way to a family name, and it sticking with them throughout their life is far from universal.
Throughout history, people have been known by a variety of names and titles during their lives. Monarchs often choose a regnal name, which may be different from their birth name.. The Battenbergs became the Mountbattens. Norma Jeane Mortenson became Marilyn Monroe. Marion Morrison became John Wayne. Karol Jozef Wojtyla became John Paul II. My wife is on her third surname, and my ex partner had documents in 4 different surnames, one of which she had made up for professional purposes.
Therefore, the fact that Jesus may have been known by another name in his early life, or throughout his life, does not cast doubt on his existence as a historical person.
However, in all the other examples above, there is clear evidence of who the person was and what name they were known by before they adopted or were given their new name or title.
It is reasonable for someone who doubts the historical existence of Jesus to seek evidence of his existence, and in order to do so, they would need to identify the name by which he was known in life, then find contemporaneous records relating to that person. This is perfectly reasonable when considering the historicity of Homer, Arthur, or Robin Hood, and it is equally reasonable when considering the historicity of Jesus.
Huge claims are made for Jesus, not only in metaphysical terms, but also in terms of what he is claimed to have said and done in his life. If we compare him to other figures of the time, we have very little contemporaneous evidence. We have none of his writings, and the reports of his preaching were written long after the event. For those who choose to believe, it is a matter of faith and I respect their right to believe. For those who prefer evidence to faith, there is less to go on than would be ideal.