There's not a single person alive who has changed their biological sex. It's in every cell of your body. There seem to be plenty who say they have changed their gender identity. So which of those 2 is innate and immutable and which is a subjective, changeable feeling....
1 I have said endlessly, trans people do not change their reproductive sex, so that is not a stick you can beat me with. A poor effort.
Biological sex is not innate. A better argument might be whether sexual identity is innate or not. I will tend to say it is not since I haven't yet heard a trans woman say that their sexual identity is female. The whole basis of what being trans is, is based on sexual identity being incongruent with gender identity (which is innate).
That I have to still be explaining this to you is of concern, that is at least to me.
2 I have heard trans people say that they gender has evolved. This can be so for at least two reasons or perhaps a medley of the two; our knowledge of ourselves develops over time, mostly in our early years, secondly in our acceptance of our selves. Educationalists have the understanding of how this happens. If this interests you, then should look at the work of Vygotsky and others. Visual and audible learning is one thing and very useful in learning some things, but not probably not so useful or understanding self. That relies upon a combination of existing self-knowledge, and new knowledge including the learning of vocabulary and concepts in order to lead to reason.
A common question is why there are so many young transitioners, another is why there are so many late onset transitioners. These are complex questions, but one likely reason that explains both is just that we live in the information age, another is that we were living in an age of greater acceptance. I say were because I believe we have experienced regression. When you consider these factors with what Vygostsky had to say about the role of language in learning, self-development, and self-control of our behaviours, then there is a worthy argument.
There's no shared identity of 'woman'. It's just being born female that makes you a woman. Unfortunately, being born female leads to being subject to certain treatment so we have single sex services, facilities, and yes sports, in order to address that.
There is a shared identity. You say you are a woman. I say I am a woman. You say you are a woman because you say you have no innate gender identity, therefore you rely on your sexual identity to assert your gender. I am not unencumbered of a gender identity. My self-knowledge is that I have a sexual identity, a gender identity, and an identity pertaining to partner preference (what some people call sexuality if you like).
Your argument relies on your lacking one component of this identity - you seem to be one short. That's too bad, and I empathise, you have a choice of 72 other genders to choose from, just pick one, or create one of your own. I don't care what you call yourself or choose how you identify, that's very much a matter for you. But I do draw the line at you insisting that there's something wrong with my brain because I have something you don't.
You don't get to rewrite my identity, or anybody else's for that matter. You've replaced innate gender identity with innate gender prejudice - now that's the result of ideology.