I have a feeling that you're talking past each other, in one big equivocation exercise.
You're talking about SELF-identity, which, yes, it's up to you. Whether you identify yourself as man, woman, neither, both, other, cat or attack helicopter, is entirely in your head. I'll cheerfully grant that there's nothing wrong with any of that. Even for Mr Boeing–Sikorsky RAH-66 Comanche over there. Good for you. Thumbs up. I'll support you all the way.
What theprestige seems to be talking about is the demand that others also identify you as that. But, see, what I identify you as, that's in MY head. And basically, sorry, you have no right to tell me what it should be, just as I have no right to tell you what the previous one should be. If I think that that guy looks like just that: a guy, and not like a helicopter, that's that, really.
I might decide to play along if you give me enough hints (like, Mr Comanche should really try wearing a propeller beanie

) but do understand that it's voluntary and not some kind of obligation.
The problem is that there's a lot of equivocation going around. And not just between those two, but also we just had a link to someone trying to say self-identity and biology are the same thing.
In fact, it's all one big equivocation and motte-and-bailey crossover, where a lot of people seem to SAY that it's about their self-identity and their right to it (a right which, again, I'll fully grant that they do have), but it ends up somehow meaning that they can:
- flat out dictate how OTHERS identify them
- turn any unrelated conversation into being about their issue
- make demands about which sports team they should be allowed on, which dorm they should sleep in, etc
And so on.
Which actually is no longer even remotely the same thing.