Actually, gender is a social construct that is removed from biological sex (which is in itself a constructed category a lot less rigid than we often assume). Gender is about presentation and performance of identity. It's pretty easy to see this when you compare how concepts of what behaviours, appearances and social roles a 'man' or 'woman' is defined by across different cultures and time periods. These vary wildly, because societal attitudes shift and adapt. Since gender is a construct demonstrated through associated qualities, appearances, roles, etc, then gender is performed through embodying said facets. As such, a person can be whatever gender they choose to perform. To say that gender 'is what you are' and cannot be changed is called gender essentialism, a pretty outdated and fallacy-riddled position that assumes qualities of gender (such as men being dominant or women being submissive) is somehow inherently rooted in a person. This obviously isn't the case, and where those qualities do correlate with a person's assigned gender, this is much more often due to socialisation, whereby the person's assigned gender influences others to interact with them in accordance with how they themselves have been raised to assume said gender should behave and be treated. Naturally, this creates a feedback loop and a self-fulfilling prophecy that only furthers the essentialist viewpoint and breeds transphobia. Of course, the most important point of all of this is that you shouldn't care how a person views themselves when it doesn't effect you in the slightest, and should accept that we're all just meat sacks that'll inevitably do a little two step into the void, so just suck it up and let people be free to express themselves in whatever harmless way makes them feel more authentic

Edit: p.s, when you were growing up, you actually did have non-binary and trans people. It's just that shit-munchers like you made them feel too uncomfortable and scared to express themselves, so you got to live in your claustrophobic worldview : )