Gender was (apologies if I've already said it) coined to mean the aspects of sex which weren't tangible, but were observable.

Consider emotions. You know "happy" means. If someone feels happy, they recognise the feeling. Some people are better than others at hiding how they feel, but overall, if someone's happy you can tell from their behaviour. People instinctively understand that interacting with someone when they're happy is different to when they're sad. People understand that some people feel happy more often that others.

It's hard to pin down what it is, but it's clearly there.

And yet, if you were to try and explain what it means to be happy, your choices are; a) "I dunno, I just feel happy" or b) absurdly complicated philosophy.

A trans person (and to a lesser extent a cis person)'s gender can be observed through euphoria or dysphoria in a similar way. We know it exists, even if we can't quite explain where it comes from.