Or is (s)he?
We can all agree that just because you feel like a man doesn't mean you want to be one. Wanting to be transgender doesn't really confirm you are transgender. If "she's not a real man", you don't believe in transgender rights. Either that, or she is a cis woman who is just having masculine thoughts.
But how do we even define masculine thoughts? Blue isn't naturally a boys color. 100 years ago, it was for girls. And you know what else is associated with boys, but isn't? Toy cars, six packs, emo and goth aesthetics, rock, punk, reggae... It's also the reason we can't define what feminine thoughts are. Pink is not naturally a boys or girls color, either. And many men and boys like skirts, Barbie dolls, etc.
So surely the concept of gender might not exist as a social construct, but it can exist in chromosomes. But I don't believe in chromosomes. If there are only 46 of them in my body, there are so few, typically, you wouldn't even believe they exist.
In conclusion, gender doesn't exist. Our brains just invented it because a p**is looks different from a v***na. So she's not a real man, and she's not a woman either, or any of the other genders we made up. She's just human, because, after all, gender does NOT, in any way, shape, or form, truly exist.