Okay, so here's the thing. Unlike someone believing they can fly, when a trans person says they are a certain gender, they don't just believe it to be true, it actually is true. Gender is not determined by genitals, or x/y chromosomes. It is determined completely independent of sex.
According to The American Heritage Stedman's Medical Dictionary, 'gender' refers to one's sex, based on reproductive organs. I mean, that's why trans people get **** changes, right? If our gender wasn't determined by our genitals, they wouldn't be having the whole gender identity crisis in the first place, would they? You could be a woman with a **** or a man with a vagina. Hell, if you're born a hermaphrodite, you could be both!
What you're saying is that gender is a matter of personal perception. If we accept that then we also have to accept that species is a matter of personal perception. A person could truly believe himself to be a peacock. As it just so happens, that's actually a thing. It's called species dysphoria.