[Got a bit long. Tldr: for identity wrt persons, read John Locke and Derek Parfit. Many trans ideologues just make mistakes. ]
It may be of interest to try to track some of the history of 'identity' in something like the way people try to use it now. Why 'identity', and what has its use for persons to do with its other (major?) use to mean the relation of being the same?
Two things are identical if they are the same. Cue lots of philosophical musings, one particular set of which is to do with so-called 'personal' identity.
A good place to start with this might be John Locke, who added a chapter in his book An Essay Concerning Human Understanding (Book 2 Chapter 27, 1694) on what people describe as the "persistence conditions" (or "conditions of diachronic identity") for persons.
This sounds terribly technical, but really it isn't at all. The question Locke and others want to try and answer is just this: "What makes me (or any person for that matter) the same person as I was yesterday ... or last week, or sixty years ago when I got married, or ... well, just what is it that makes me be the same ('identical') person as I always was in the past and will be in the future, even as I change various aspects of my body and personality, grow old and so on?"
This question - what makes me the same person over time - might be glossed as, 'What is it that makes me who I am? - What is it that makes me, me?"
This is the philosophical problem of personal identity. Of course it has practical aspects as well as sheer intellectual curiosity driving it. Is that old person, Fritz, the same person as that young man who was a concentration camp guard in 1944? How do we decide? And so what? ... Am I really the same person as when I was a religious believer, all those years ago? ... Is this person, Claudette, the same person as Claude, who you say attacked you that evening? ... and so on.
These are questions of personal identity. Interesting, important questions, with ethical and moral import as well as metaphysical (and logico-semantic and so on, perhaps depending on your philosophical bent).
OK, now the idea of 'identity' in this sense has undergone a bit of slippage. So we talk about ID ('Identity') cards, for instance - cards we can use to identify ourselves, to say who we are, forgetting the origin of the term in the sense of 'what makes me the same over time'. And so on.
And 'identity politics'? Sometimes consciously, other times not, people having no real answer to Locke's question encourage this slippage into talk of what makes a person be the person he or she is, independently of any reference to the kind of diachronous identity (of what makes one be the same over time) Locke was talking about.
What is it that makes me the person I am? Lots of answers can be offered. But, anyway, that is what my identity is, in this sense.
Of course what many people (amongst which almost all trans people and their apologists) mistake about this is the possibility of error in self-ascription of this sense of identity. For most X, the claim "I identify as X," as it turns out, is no guarantee, even if sincerely offered, that I am even X at all, much less that X is a persistence condition for my personhood.
[Oh, and that spinning sound you hear? That comes from John Locke's grave.]
[More seriously, if you want some more recent material about personal identity, try Derek Parfit's Reasons and Persons (1984). I don't say Parfit gets everything right, but he's always worth reading. Dead now, of course. But not as long ago as Locke.]