How come you believe gender identity is not a real thing?

I think it's a real thing for those who say they have one (their mind, their experience), for the rest of us, no we don't have gender identities.

Do you think it's false that trans people have the brain of their opposite sex?

Yes.

Where is your evidence that trans people don't have the brain of their opposite sex in them?

As this study puts it "The present data do not support the notion that brains of MtF-TR are feminized". Badly designed studies may show brain similarities between trans people and the opposite sex when they don't correct for sexual orientation. When it is corrected for then "data show that whereas homosexuality is linked to cerebral sex dimorphism, gender dysphoria primarily involves cerebral networks mediating self–body perception".

Are there any disadvantages that come with male and female being used to refer to other things besides sex?

It leads to people being gendered against their will just for being born a certain sex. If you want a word for gender indentity create a new word for it that isn't an existing word for sex so people are not associated with a gender identity just for being female or male.

Thoughts on xenogenders and neopronouns? Are they real things?

I wouldn't have heard of neopronouns if they weren't a real thing. People can invent identities that are real to them, and I don't mind what people do in their heards but I'm not going participate in their role-playing.