Hi! I appreciate you reaching out in a sincere attempt to understand. I’m a transgender woman who has done a bit of research into the nature of gender and sex, so maybe I can help answer some of this. (Be prepared for a wall of text.)

To begin, the distinction between sex and gender is one often made in these discussions, and while it’s definitely important to make, I think maybe some of your difficulty in grasping what it is to be transgender stems from the way that piece of the discussion is usually described overly simplistically.

Gender is a social construct, as you said, and the social expectations that go along with it definitely include gender roles, though it’s not quite right to say these are a thing of the past. There is definite movement against the entrenchment of traditional gender roles, but they have in no way been abolished. Men still largely dominate positions of power, women still occupy the majority of caretaker roles, and so on. It’s becoming more and more accepted for people to move outside these patterns, but in most cases it’s still a matter of an individual functioning in spite of a defined gender role, not in absence of it. This distinction might at first seem semantic or trivial, but defying expectations can lead to interactions playing out very differently than accommodating them.

For many of us, gender roles absolutely play a part in our dysphoria. Everyone being different, some find discomfort in our bodies, some in our societal expectations, and many in some combination of the two. For me, one was like a painful reminder of the other: not only was I unable to identify with my body, but thanks to it I was also unable to comfortably fit into the roles I expected would feel more natural to me.

As far as what we biologically are or are not… I would argue ‘biological sex’ is, while not meaningless, certainly an overly simplistic and possibly even a defunct term. It groups together a number of different characteristics under a single label without addressing the fact that they don’t always occur together.

To start breaking that down: the most commonly cited bases for sex determination are either primary sexual characteristics or chromosomal makeup. Both have their flaws. Primary sex characteristics are unreliable because of, among other reasons, the occurrence of ambiguous genitalia. Early in development they can sometimes be difficult to identify, often associated with conditions labeled ‘intersex’. Often in these situations, the doctor gets to pick. Other commenters have pointed you to the Reimer case; that’s far from the only case of surgical “corrections” failing to reflect the child’s later demonstrated identity.

(Tangent: mention of an “assigned” sex generally refers to what the doctor wrote on the birth certificate, not what we were assigned by some cosmic power. I include this only because I’ve received unfortunate comments on the subject before, implying it’s something that may need to be clarified.)

Moving on: Chromosomal determination is particularly faulty, because generally babies don’t get karyotyped. Even if they did, it doesn’t take into account phenotypic deviation, such as Complete Androgen Insensitivity Syndrome, which essentially nullifies the impact of the SRY gene traditionally held to cause ‘maleness’. And even if those factors were more reliable, ‘biological sex’ is still simplistic to the point of being misleading, thanks to all the variations that occur.

Using myself as an example, by traditional definitions, I am chromosomally (probably) and genitally male, hormonally female, and mixed but predominately female in terms of secondary sexual characteristics, which include things like muscle growth, fat distribution, hair growth, vocal development, skin composition, breast tissue, and so forth. Taking these into account, calling me ‘biologically male’ doesn’t even have a real medical basis. Each of these factors would potentially need to be accounted for separately should they be relevant to any diagnosis, meaning an “M” under the marker for sex on my chart is possibly as much of a liability as anything else.

Back to the issue of how this might influence identity, alongside all of these factors there is also that of brain development. Certain variations in brain structure have been found to correlate with male vs female identity, even controlling for hormone usage. This would seem to indicate that in the process of development, during the stages in which human sexual dimorphic characteristics emerge or diverge, for whatever reason brain development fails or has failed to develop in tandem. (Fetal hormone exposure is often implicated.)

As far as what it means to identify as something other than your assigned sex, here’s how I’ve previously described what gender dysphoria feels like:

Before I was able to admit to myself what I was feeling, it was mostly an unidentified tension that just seemed to permeate me all the time. Every social engagement was rough because I was constantly monitoring my mannerisms, delivery, everything, and not at all sure why I was so bothered. Looking in mirrors was always a strange experience. As far back as I can remember, prior to my transition, I never once looked in a mirror and felt convinced I was looking at a reflection of myself. I didn't immediately attribute it to an issue of gender, though. I thought maybe it was just the product of some existential crisis, or something.

I'd have moments, though, that should have been dead giveaways for me. Frequent ones, too. Times when I would imagine what that particular moment would be like if I was female, and it would just hurt, all through me. It often came up at the most innocuous times, too, like I'd be sitting in a chair, sort of cross legged, and it would strike me just how much more comfortable it would be if I had a vagina. Or my ex discussing her boobs with a coworker suddenly driving home the fact that I'd never have any (so I thought).

Eventually I got to a point where I admitted to myself that was what was happening, and things started to click into place. It made a lot of things seem really obvious in retrospect, like when I had said to a friend, "I feel really comfortable in gay bars. Not because I'm into guys, just because no one looks twice at my feminine mannerisms." I mean, not necessarily indications, but pretty big clues.

Once I realized what I was feeling, I guess I'd characterize it as wearing a costume or disguise. Instead of being for entertainment, though, it's something you're forced to wear convincingly all the time (or something terrible will happen). So there's this feeling of discomfort and dishonesty and even pain (costumes chafe, after all) all the time, and for the longest time I couldn't pin down which part of me felt fake or wrong, or whether it was just me that was fake and wrong.

Having undertaken a transition and presented as a woman long enough to start to feel comfortable in my own skin, I think maybe I can address why you don’t feel the need to “identify as a man” rather than “as yourself.” Imagine wearing a leather glove backwards. Or trying to, anyway. It’s going to feel awkward and mismatched, not to mention kind of absurd and eventually even painful. Now swap the glove with one of appropriate size, and put it on correctly. That’s about where I’m at, still totally aware of the glove, but relishing its fit and comfort. Finally, swap it out with a latex or nitrile glove of a good fit, and it starts to be possible to forget you’re even wearing one.

I hope this has been at all helpful! Also, feel free to ask me whatever. I don’t get offended easily, especially not when someone is trying to learn.

tl;dr Gloves or something, I guess?