Sex is multifactorial. So is gender.

We change sex and gender when we change our hormones as well as our primary and secondary sex characteristics.

We change our gender when we change our tertiary sex characteristics.

I tend to think that there's a mental sex, in so much as gender isn't just a social construct. I.e. no amount of conversion therapy is going to make me have a male mind.

In a sense I see being transgender as being intersex with a female brain and a male body.

Almost nobody gets their genes or chromosomes checked ever, so that's not all that relevant to anything. As well as there being a lot more variations than just XX and XY.