Okay, as I understand it mostly no and sort of yes. It’s primarily due to the social element; gender is a socially realized phenomena that only ‘exists’ within each individual. For a Trans man, his identity has been realized to himself at the individual level this is fundamental for gender identity; however, because gender is socially created, they are only socially recognized as a man if the people around them do so. This will put them at odds with not only Patriarchy but capitalism (capitalistic production relies on reproductive labor of women for the purpose of more labor i.e. for capitalists, women exist to birth and raise workers) and, more often than not, bar them from society at large, subject them to violence, etc. Trans men want to and could perform the typically masculine roles in society (thereby becoming men or if we’re being overly pedantic then possibly an adjacent masculine gender that is for all intents and purposes a man) you have certain societies that aren’t a binary, such as having two masculine and two feminine genders, (see the Bugi people of Indonesia for prime example) that reinforce that the patriarchal binary is certainly bunk. Cis men have a vested interest in maintaining patriarchy so will tear down Trans men (and any queer person) as it disrupts the binary/their position, a portion of women have also sided with patriarchy/reaction for reasons that would take some getting into. As gender is socially realized, it can be pretty nebulous to pin down, but I think approaching it through a class lens as many feminist and queer theorists have is a pretty solid foundation you just have to be wary of terfs (as with many things radically feminist).

That probably didn’t provide a satisfying answer for finding a definition, but I personally don’t think there is a satisfying answer. It’s possible there is one out there, but I just think women are women who say they are women and men the same, society has just barred some of them from performing their preferred set of reproductive labor.

this is the ‘me rambling a little bit before I go to sleep’ edit: I think it’s also important to re-emphasize that the definitions I provided were for Patriarchy, i.e. how men and women are expected to operate under patriarchy and often do but not 100% of the time when living in said system. Such definitions don’t operate so cleanly once different class oppressions meet. A bourgeois woman probably doesn’t perform significant reproductive labor as they can afford to hire a nanny, tutor, surrogate, etc. but she almost certainly has a different set of gendered expectations that would be socially recognized as typically feminine. So an analysis of gender also requires analysis of race, economic position, disability, and so on to truly understand the ‘definition’ of one’s gender.