I think you fundamentally disagree with this, but I'd say gender is your internal sense of how you align with societal constructs that we associate with certain "gender roles". In our society this is "male" and "female", but other societies have done different things here with the same underlying biology.
Trans people are, and I'm generalizing here, people who feel their biology doesn't match up with the gender role they identify with. They often then want to align their physical presentation with that associated with the gender role, on the belief that societal roles are more important than biology. (It's very transhumanist, in a sense.)
I'll note a fairly easy example of sex and gender differing even in our society, which is intersex people. I.e. those whose physical expression lies somewhere between male or female (which isn't super-common, but certainly happens). They're ambiguous physically, and we historically make them pick (or pick for them at birth) what gender role they'll perform.