Since this conversation is being had still, then I will say something on it.
Trans people do realize they are not real men or women biologically. They can not help the chemical imbalance in their brain and for their health need to transition. Pretty well understood at this point.
The gender identity ideas didn’t really start happening till the past 10 years or so. And I will admit has gotten a little ridiculous on some although I understand the premise.
Yes, there have been trans people all the way back to Egyptian era and then some, but it was more taboo back then and not spoken of as much. Even in the time of Shakespeare there were gays and trans people, but so highly frowned upon it was typically kept a secret. Do I agree with this? No.
Nowadays, surgeries can be done to make you completely the opposite sex without leaving a scar and you honestly can’t tell the difference. Hormones help maintain this appearance a bit due to once the genitals are removed and replaced your body no longer naturally creates hormones. But, nowadays you can fully transition, hormones, surgeries, name changes, gender legally changed without saying a word and no one would know the difference.
As for medically, yes you have to state you are trans so they can treat you properly, but otherwise this point you have the parts of your chosen gender and medically treated as that gender otherwise.
But, case and point Trans people don’t typically say they are real men or women unless they are extremists. Most simply state I am a Trans man or Trans woman. And ask to respect their pronouns especially if they look like a woman or male at that point. In early stages typically they except being misgendered because they do not look like the gender they are trying to perceive. Otherwise, most are decent people. Biggest thing they hate is being treated like a fetish or they aren’t a person. The little trivial things they typically ignore. And if someone can’t respect them, then they just won’t be around them.
I have had plenty of trans friends and work with a few. So, I have had some of these conversations with them and done my research so I can understand better. Will everyone do that? No. But, I would rather understand and see how something ticks because I enjoy learning something new and crave knowledge. Even if its rather trivial.