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 sex 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 penis 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.