Trans people know that even if they transition they will not be biologically transformed into the sex they are transitioning to.
"I'm the female gender, not the female sex" makes just about as much sense as "I'm the individual, not the person."
I already explained the difference between gender and sex in my post. It is a very real difference that society is finally coming to terms with and why jokes like this were completely acceptable at the time and now may be a bit removed from where society has progressed to.
Though even with that in mind, transgenderism as a concept is actually much older with there being many instances of transvestites or transsexuals existing in past centuries. I would recommend watching Ed Wood's 'Glen or Glenda.' It's not a good movie at all but is fascinating in how ahead of its time it is and how in explaining trans issues almost makes it resemble a film today (maybe minus how bats*** crazy it gets at times) even as it was made in the 1950's.
I don't want to sound like I am speaking for the trans community though, some of whom post on this website. I would be curious to what they would have to say about this scene from the film.