Emily's Cat
Rarely prone to hissy-fits
Have to be honest and in reality there is a stereotype of women and men that society still expects and that includes outward presentation, this also includes behaviour but we don't need a dissertation! Sure some women and some men don't go along with that stereotype and many women and men decry its existence, but it does exist and most women and men fall into it. I would call that stereotype "gender", I consider akin to DNA and the phenotype - in this analogy the DNA is the biological sex, the phenotype is the gender. With that in mind it I think it is accurate to say that trans men are woman that take on the gender of a man.
I could quibble with your analogy, but I think the gist is spot on.
There are some behavioral tendencies that are evolved as a result of our sexual dimorphism. That shouldn't surprise anyone, we observe behavioral differences by sex in many other species, and presumably this is influenced by the disparate burden of reproduction. We also observe pretty regularly that these are tendencies, not dictates. Lionesses tend to do the hunting, Lions tend to do the fighting (almost always against other males). But lions can and do hunt when the need arises, or when they're solitary. And lionesses absolutely can fight off attention from an unwanted male or an aggressive wildebeest.
There are going to be innate tendencies in humans that differ by sex. It would be irrational to pretend that none exist (and yes, some of my predecessor feminists are irrational on this topic).
But as you alluded, there's also a social element that is enforced by culture, and isn't necessarily correlated with the evolved behavioral tendencies. Those are the norms and expectations that show the most variation across culture and era. Fashion is at the top of that list. That females in western society are expected to have longer hair and wear make-up isn't an evolved thing, it's social in nature, and it hasn't always been the case. That males are disallowed to wear dresses or frills is equally external - there are definitely times where males have gone in for a lot more adornment than females have done.