Here's my take on it... based on what articles there are out there and my impressions of those articles... it's opinion - that's just where mine is based.
I understand that there is a part of the brain that makes us male or female. I personally, as a transgender woman pre-HRT, consider myself to always have been female as I believe that is the most important part of gender. I consider myself to have been born with an intersex condition - though I'm sure current medical standards (and insurance plans) wouldn't agree. The wrong hormones make this condition painful. It is where the body dysphoria comes from. Our rooted understanding of our own physical gender is hormonal, but the knowledge that we're of one gender or the other - our gender identity - is intrinsic (non-hormonal). I think this is why many of us are more strongly transgender at puberty than early in life... the understanding of our gender identity (non-hormonal) can be overridden by a strong societal gender imprinting. Society's imprint often gets overridden when the hormone flooding overwhelms the gendered part of our brain, causing (often extreme) anguish and discomfort. Those of us who pass through puberty without coming out still have the anguish, lessened by the end of puberty. We may learn to deal with it in other ways... but we are still female, even while trying to deal with appearing male. We may not know that we're female.
I also understand that most gender roles are imposed by society. Girl's toys and boy's toys are equal to the child raised without imposed gender roles (something I think is practically impossible in current American society, though it can be done incompletely). These roles are imprinted by age 3 and strengthened throughout our lives. I believe there are intrinsic qualities to being male and female, which a recent article suggests is not true, but in my experience (and in talking with other trans women) we communicate better with women and tend toward more feminine activities (though the latter is hardly set in stone). Male and female activities are like boy and girl toys at the adult level. There's even more crossover and activities dominated by one gender or the other is primarily societal, though partly physiological (sports vs. reading, for example - one is active, the other inactive - both men and women often enjoy both activities).
I think we usually fit in with our true gender, not our assigned gender. This cannot be entirely based on societal imprint of gender. If all women loved to talk about sports I don't see how I'd be interested in sports now. I could very well be underestimating society's role on our own gender identities, but I have to believe there's a deeper reason I don't usually enjoy male conversations.
We're female, but that also doesn't make us anything we don't want to be. Every woman is different. There's no such thing as "girl-this" or "boy-that," something I'm trying to teach my own kids now. If I want to drink beer and watch sports, it doesn't make me less of a woman (I know lots of women who do). If I don't care about my appearance and want to believe that society's requirement for women to look good is wrong - that's fine... but if I want to wear make-up and look as good as I can (within reason) that's ok too.
I've gotten accused (veeery passive aggressively) by my wife of being vain. I think we, as trans women, have a tendency to act more feminine, with make-up and clothing, because we hate being so masculine in body. There's nothing wrong with that - though it's probably bad to take it too far. Personally, I haven't - but my wife has never used much make-up (I don't think she owns any lipstick) - so to her it looks like I've gone off the deep end with it.
I'm pre-HRT, so my views may change significantly (I only have others' stories to go off of), but I don't believe any of what we do is required to be female. I think our problem is about half seated in hormones and physical body and about half in societal impositions (give or take). We like to feel more feminine because we've been further from feminine than any other woman for all of our pre-transition lives. It's not required, but it feels good and helps our problem... and that makes it right for us to do, but that is not what being a girl is.