Similar to Hogwarts, there is a school for magic practitioners which could only be accessed via invitation. To graduate the student must accumulate enough credits throughout the coursework and personalize their own wand or broom for ladies, however for the crafting the student had to seal their name and bind it to the wand. The consequences is perfect amnesia for the user and rest of the world, so that any memory and traces will be replaced by another name automatically. Should the wizard wish to abandon their wand they can do so only if they could speak of their real name and the process is irreversible, tossing away the wand is the same as becoming a Muggle. My question is why should the rite of passage for a wizard involves identity crisis literally?
"My question is why should the rite of passage for a wizard involve an identity crisis literally."
The wizarding worldview is radically different from the mundane worldview. Throwing your world into question necessarily throws your self-concept into question since we are a part of the world and understand ourselves as somehow produced by it, living in it, and interacting with it (except for the solipsist, for whom it should be even more intuitively clear why an adjustment in worldview would necessitate an adjustment in self-identity.) Even moving from one country to another, or from one city to another, can leave you scrambling for a place in the world on which to re-erect your sense of self, and when you find it you typically find your former self doesn't fit. If you're the type of person to stay in one place for decades, you have the luxury of adjusting yourself little by little so you barely notice the change in yourself corresponding to the spirit of the time, assuming you maintain active engagement with society. Otherwise, you'll wake up one day finding that the world has left you behind and you'll either further isolate to preserve your sense of self or have a crisis of identity as you attempt to re-engage.
I submit that the transition from a mundane existence to a magical one is at least as disruptive to one's sense of place in the world (and therefore identity) as a transition to living somewhere with a local language you've never spoken. That's why you get a crisis of identity, there's simply no way of avoiding it.
If you consider that magic in literature is frequently a metaphor (transparent or otherwise) for the creative power of art (whether because authors have shamanistic ideas about art or because they're very, very up their own arses as authors), the crisis of identity (or at least, a history of having had one, so you know how to travel the path) is necessary to think that which has not been thought. A good author, to write believable characters whose behavior makes any sense to readers, must be able to step out of his or her own identity and into another and consider the world from that perspective. When authors fail to do this, you'll find people being totally turned off. Just think of it: you're reading the book, and the character you like best who you think is doing everything right and believes all the good things (the one you identify with) just up and does something so boneheadedly idiotic that it completely breaks your ability to conceive of them as a coherent entity. WHY would anyone with that worldview behave like that? Like when Vladimir Lenin lived and died in a mansion. (Oh, wait, that actually happened.)
OP wrote in comments:
why any talent would choose birthright over status.
To give up family, friends, etc., the profession of wizard is going to need such respect and authority to make it worth it... or else all of your wizards are of low-class birth, where the sacrifice is not so great.
You might have a caste of untouchables, the amnesia process becomes a cleansing rite that allows the person to be a part of higher society.
If high-born and wealthy aspire to wizardry, that implies wizards have a status to make it worth it.
Fun idea: let the wizard's mother or father retain the memory... they and they alone can speak the name that ends a wizard's power. The wizard doesn't remember parents... so the wizards have a saying, "Would your parents approve?" with real ominous overtones.
Names-based magic has been treated in books like the ones of Earthsea and The Kingkiller Chronicle. In those worlds, everything has a real name which can be used by powered people to control the thing.
In your world, the existence could be related with the real name of the elements that form it. This way, the world is a protection of the name's reality, which creates the physical rules.The names that humans give themselves are not their real names, and therefor they don't have power over reality.
But long ago, a man whose real name was especial got in contact with the name's reality. He discovered the way of carrying a human name to the other reality and make it the real one. By doing so, the owner of the now real name would be able to interact with the other real names and change the reality; this has been known since then as magic.
During the ritual, the wand, whose real name is known by the masters, works as a bridge by which the human name of the apprentice travels to the name's reality. When doing so, the name is purged from the human's world. And, as the human name is a false real name, if at anytime it was said, it would be expelled forever from the name's reality.