It's interesting that all of the oldest vampires we've seen in VnC so far, all those that have had the time to feel the weight of semi-immortality, seem to somehow be divorced from their "true selves."
Teacher's whole gimmick is that he's an ever-changing nameless shapeshifter. His name and appearance are whatever suit his mood on a given day, "true self" be damned. And seeing as even Machina doesn't know what his deal really is, he's apparently been living this way for a very, very long time. Vampirism for him is inconstancy.
Machina has also spent centuries publicly divorced from her "true face," though she does at least possess a true face in a way that Teacher doesn't seem to. It's just that her face identifies her as a member of a group who were killed to the (almost) last for fear of their power, so she's kept herself hidden away for centuries. It's unclear how many people know who she really is, but if her colleagues in the Senate don't know (and they may not, since she wore a veil to the Senate meeting), then she's been living this way since the very beginning of vampire society. People only know her through a series of disguises, theatrical suits to hide her face and skin, and it seems like she may have falsified her name and/or gender as well.
Faustina is the ultimate identity-less vampire, living as a twisted echo of herself in the form of a malnomen. She is a hidden corpse. She is an ever-changing formless shadow in search of the names of others to steal. She cannot ever be allowed to remember her name, who she truly is, lest she become even more of a mad nightmare tyrant than she already is.
Whatever's going on with her, the severing of her power and being from all knowledge of who she is seems to be an integral part of it.
Even Luna, despite being a different sort of vampire with a harder-to-pin-down age, falls victim to this lack of identity. When Vanitas asks them what they are in mémoire 51, they respond that they've always wondered the same thing, but they regret trying to find out. Luna lacks an understanding not just of who they are as an individual, but what sort of being they are to begin with.
(Or at least, they lacked that understanding at some point. It's possible part of their regret when Vanitas asks may come from finding unpleasant answers to some of their questions. But regretting and fearing the truth of what you are would still hardly be a "normal" relationship to self-understanding and identity.)
Not to mention, perhaps more importantly, Luna did not have a name for the vast majority of their long lifespan. They had things people called them—Vanitas, The Vampire of the Blue Moon—but they never had a name they identified with until they took the name Luna from Vanitas.
Chloé is the only extremely old, powerful vampire we've met so far who doesn't seem to have this disconnect from her "true self," but I think you could also argue that part of the point of Chloé's character is that she hasn't felt the weight and press of time in the same way as the other oldest characters. She's centuries old, but she's spent the vast, vast majority of her years in stasis.
Chloé spent the ~130 years between the first beast of Gévaudan incident and her salvation by Vanitas living in a time loop. Her malomen created a self-contained infinitely looping world for her to live in, untouched by the outside and without growth or change. And even before that! She spent her whole earlier life in a state of waiting, waiting for her family to finish their construction of the formula alteration device and "fix" her so her life could resume. She was frozen in time at age four and not allowed to progress until she was rewritten and human again. That's why her arc has the winter theme; she's literally frozen in time and not allowed to progress or feel the change of seasons. That's why, symbolically and perhaps literally, she stopped aging at the age of eleven.
The only vampire we've seen who's lived for several centuries and isn't somehow divorced from herself is the one who was frozen in place as a child and never allowed to become herself in the first place.
Machina in Mémoire 65 states that boredom is the most terrible ailment there is among vampires that are centuries old, but that's not the only ailment threatens them. Teacher in 62 says that, among vampires who have lived "too long," shifts in character are expected. There is a certain degree of unmooring that happens to anyone who lives for hundreds of years. You get bored; you've seen it all, and the person you are begins to shift and warp with the passage of time.
Two of the greatest looming horrors in VnC are the threat of the distortion of the self and the threat of the corruption of death. The fear of the distortion/loss of self is the chief horror of Naenia and her malnomens—the main threat/villain that drive the entire story. To be changed into something other than yourself is the ultimate loss of agency, a loss of control over not only your body but your very being. There can be no greater violation than the rewriting of one's true name.
Meanwhile, the inherent horror of the corruption of death is something that has been brewing in the background of the series for a while now. It's hinted at in all the yet-to-be-revealed plot threads in both the Ruthven and Faustina storyline and the Mikhail, Teacher, and Luna storyline. It's there in Vanitas's absolute insistence that resurrection is impossible, that a resurrected person would not truly be that person in the ways that matter. Death in this series sometimes serves as an escape valve, a way of reclaiming agency for people like Vanitas and Jeanne who have no other way of controlling their bodies or their circumstances. They rely on the promise of a death they choose to cope with the fear of a loss of their very selves. To tell someone they have no right to die, as Charles does to Astolfo, or to attempt to resurrect someone, as Mikhail does to Luna, is monstrous, because it unmakes the freedom and certainty of death.
This whole series is built upon the idea of Vanitas, the inevitability of death that renders all else meaningless, and over and over we are told that nothing good can come from attempts to deny the inevitable.
Vanitas the concept tells us that death will come no matter what, and Vanitas (no Carte) the story tells us that this inevitability is better than the alternatives. That's why so much of the story is about Noé, our narrator and POV character, having to come to terms with the fact that death cannot be avoided. Death is frightening, but to take it away would be infinitely worse.
Given all that context, then, I think it makes absolute sense for the story to examine what happens when somebody doesn't die.
What happens when somebody lives for centuries, when they have the potential to live forever? What does it do to them psychologically?
On a thematic level, if death's inevitability is good, and if death provides a way for desperate people to retain the self they would otherwise lose, then of course the inverse is true. Of course eternal life heralds some form of loss of self. Teacher, Luna, and Naenia's namelessness might not be caused by their old age, nor is Machina's need to live in hiding, nor is Naenia and Teacher's physical instability, but I don't think it's a coincidence that we're yet to meet a single elder vampire with a stable relationship to their own identity.
Time warps people; Teacher directly says as much to Machina. And all of the above is why he discusses people who have lived "too long," rather than simply people who have lived a long time. The denial of death is unnatural. It is dangerous, foolish, and inherently a horror. And immortal life is one extremely long, slow denial of death.
It follows, then, that if death provides the escape from the warping of identity, the denial of death inherent to immortality must create a long, slow, warping of one's self.
