Gender is a nature vs. nurture debate, some parts of gender identity are biologically determined, others socially determined. That's how identity is formed.

Self-IDing as the opposite sex is following an entirely different view, one that rejects the scientifically sanctioned nature vs. nurture debate, in favour of "individual freedom of choice". It's the idea that you can independently determine your own identity. This is the self-ID cult's view of gender, which is actually perfectly concordant with libertarianism.

Sex & the resulting socialisation into the gender norms associated with it form identity, which is why a 'cross-sex identity" cannot exist. It's just role-play