Not true - if you go on social media such as Twitter and Reddit, you'll often see trans people making the claim that they've actually changed sex. Along with other nonsense, like transwomen claiming that they have a menstrual cycle.

I can find you people of any ideological stripe who will say all sorts of outlandish and astonishingly wrong things; this does not mean that you should choose those worst examples as your exemplars. If I did that, I would have to assume that anyone right wing was a full-on nazi, which even with the nascent fascist jock itch we're all suffering, is ridiculous.

But the fact is that Sex and Gender are two different concepts - one, as you point out, is a biological matter having to do with reproduction, and is settled and done, usually, long before the body becomes conscious. Gender, on the other hand, manifests mostly as performance, but nonetheless arises from a deep-seated place in the mind. So if someone says their gender does not match their sex, this is not a difficult concept to hold in one's head.

I don't doubt you've seen people saying horrible things of all kinds on Twitter and Reddit, but understand, those places have nothing to do with the real world. You're here because on some level, you do understand that, and you want better. Welcome.

These are surprisingly common examples. The idea behind the first is to redefine sex as a collection of attributes, and then claim that because a significant number of these attributes have changed, then sex has too. So a transwoman who believes this may point to his estrogen levels, breast growth, etc. as an indicator that he is female. This is why many claim to have a menstrual cycle too, it's intended as further indication of being female.

When people say that their gender doesn't match their sex, they're expressing a rejection of their material existence. These are men who don't want to be men, and women who don't want to be women. I think it's no surprise that many then engage in self-deception, looking for ways to support a belief that they aren't the sex that they actually are.