((There are a lot of things that could be described similarly, but I personally hold to the general "identity and identification" thing. For instance. I have a PC at work, which used to be my home PC. Over the years, stuff broke down and was changed out so much that the only things possibly still remaining of the original PC tower are some of the screws, wires, and cooling fans. It never ceased to be my PC. Look at it then and look at it now, and the difference will be apparent, but so long as it is still identified, collectively, as my PC, it remains that. If I were to buy a new PC tower and transfer some of those old screws, wires, and fans into it, the proportion of the old PC remaining in the new one would be the same, yet it would no longer be my PC, as the new PC is a new identity, and adding bits of the old one does not change it.
Similarly with changes in age, time of day, temperature, etc. Look now, and look later, and unless the magnitude of change isn't significant enough, the identity will remain. There is a fuzzy barrier between definitions of "morning" and "day", for instance. If you keep checking the time of day by the second, at some point you will become certain that it's day, because it can't possibly still remain morning, yet the exact point will be determined by your criteria and your perception of the meaning of "morning" or "day". Same with an aging person, there is a threshold to cross until the new identity becomes sufficiently prevalent, but until then a person who's always been seeing a growing up man as a boy, will refer to him as a boy, and perhaps other people who haven't been growing accustomed to the changes will see a "man" rather than a "boy". It's entirely a question of identity and identification - and like I said, something of an ontological one. In your eyes, an object that does not significantly change between the times you observe it, tends to remain the same object.
It's seen quite a few times in real life, and in anime, and everywhere really. An example I can bring up is the Alt Eisen from Super Robot Wars OG. It can be blasted to bits over the course of the campaign several times, and it's constantly being repaired so much that it might as well not have any single part of its original structure intact by the middle of the game. But after a single instance of massive damage that nearly destroys it, it needs to be rebuilt. Even if it's still keeping more parts of it than remained of the original Alt Eisen after so many repairs, but now it's redesigned and sufficiently different from what it was, before the event, that it needs a new identity.
Following this line of thought, it's pretty clear that the body or the DNA is not something that determines a "person", at least in the ER-verse. There is no bodily change more drastic than having everything but the brain replaced with robotics, or for instance what Xan did to himself, yet the characters still remained themselves after such things.
What happened with Elizas was, however, closer to the mark - a combination of drastic bodily damage/transformation and psychological pressure/trauma, on top of existing mental instability, pushed him/her towards a complete character change. Well, almost complete. Still, it was a sufficient change in both body and mind that even the character him/herself came to identify as someone else.
That, I think, provides a better example to my opinion. It takes a
gradient of perceived change, to change the identity of an object or person. Much like you wouldn't notice a person growing up if you live with them, yet would immediately notice the difference if you were separated for a few years.))
((ninja edit: Yes, that was also an example, though it provides a more... eh, technological tangent on the problem. With a robot, the artificial immortal brain is a rather obvious point of difference, and indeed what in-universe "defines" a robot, so it's perhaps not as relevant as it could be. Still, an interesting read. (or watch, if you're more into movies)))
((edit2: also, belated wall of text warning. Boring day, thoughts pile up.
))