OK, so.
Trans people are humans and as such have human rights.
They do not have the right to be right, even about themselves. Humans believe all sorts of things about themselves that turn out to be wrong when tested against reality. Believing sincerely that I am a brilliant scientist does not deliver me a physics PhD or a professorship. Even though science ability is a thing that really could be viewed as a spectrum.
Gender identity means different things in different contexts but broadly it is stories we tell ourselves about how we fit into social constructs. Stories we tell ourselves are heavily influenced by what the social construct actually says, culture, stereotypes, peers and teachers in younger life. They can change over time.
However, because of the dependence of this concept on stereotypes, which everyone knows are problematic, gender campaigners have gone further to posit an entirely independent culture independent innate unchangeable "gender identity". Thus the old sex is overwritten by appearance and stereptype preference turns into sex is overwritten by gendered souls as manifested in whatever someone says about themselves.
It doesn't make sense as an argument as children must somehow learn which label to put on their gendered soul, and the only available mechanism is stereotypes and appearance.
Gender campaigners switch at will between stereotypes-and-appearance and gendered-souls, depending on which they think will make a given argument stronger. So we can't challenge anyone in the changing room because gendered-souls. But we have to sort by gender not sex because stereotype-adoption (picture of convincingly passing TM).
And actually, we don't. Neither gendered-souls nor stereotypes-and-appearance are the reason for segregating certain things by sex, the differences between sexed bodies are. Even if some concept of brain-sex could be demonstrated and tested for, the 100m dash should still be segregated by sexed bodies, because sexed body differences are the reason there is a women's category in the first place.