Trimming for length...
Like you I get confused about what the difference between 'wrong gender role' and 'wrong gender identity' would feel like. Note that I'm not saying or asking what the difference is (in, for example neurobiological terms). I'm (temporarily) not asking for two separate explanations, at this point. I'm assuming there are different underlying explanations. Krypton Iodine Sulphur touched on them.
Honestly, I remain skeptical on this point. I am perfectly happy to treat trans people with respect and dignity and use whatever pronouns are requested. I just can't really manage to get my mind over the hurdle wherein the only people who are capable of experiencing what it feels like to be a woman inside, in a way that is separated from biology or social gender roles... are all people who are males. That strikes me as having questionable validity as an entire concept. If feeling like a woman on the inside is a legitimate thing that isn't related to reproductive sex, body type, etc. and is not a simple result of how society treats a person based on gender roles, then it seems like people who are female women and have always been should be capable of feeling that same thing and expressing it somewhat.
My limited, specific question here, to try to untangle roles from identities (hey I'm sure they are interactive so I'm not looking for complete separation) is only, what is it that leads, let's say a young person, to feel that their body is the wrong one? I freely admit that I do not know enough about transgender and that that question might therefore seem dumb to someone who does.
You can see what I'm trying to untangle here. In the hypothetical complete absence of society, and thus gender roles, would such a young person still feel they were in the wrong body?
My guess is yes.
On this, I have an entirely unpopular opinion, which is highly likely to piss off a bunch of people.
Being unhappy with one's body or looks is a known thing. It happens. It happens to almost everyone to a very mild degree. I don't like my hair, and I wish it were thicker and fuller and I didn't have a five-head. I wish I had nice strong fingernails that didn't delaminate constantly, and didn't grow crooked. It's very common to experience this kind of dissatisfaction and discomfort during puberty, when one's body is changing. It's natural to compare oneself to those around one, and to wish that one looked more like this person or that.
For some people, it's a bit worse. They really strongly dislike or are uncomfortable with how they look. They feel like the flaw is wrong, it makes them self-conscious and anxious. Some people have clear reason for their feelings, if they are far outside the norm, or have an obvious divergence from expected human conformation (being cross-eyed for example, or having a speech impediment). For some people, it's not so clear... but it still happens. Some people resort to surgery to augment their bodies or faces to more closely resemble how they feel it should be. Some fall into anorexia or bulimia in an effort to force their body to look like their idealized internal view of themselves.
I wouldn't treat someone who got fake boobs as a pariah. I wouldn't abuse or harass someone dealing with body dismorphia. I would do my level best to accommodate and support them instead.
On the other hand, however, I wouldn't accept the position that anorexia is "normal human variation". And I wouldn't accept as true the claim that someone was born into a body with the wrong boob size.
Brains are surprisingly plastic. The things that we spend a lot of time thinking about, especially if we attach strong emotion to them, change the wiring in our brains.
Try the experiment. Over the course of the next week, when you get up in the morning, look at yourself in the mirror naked. Tell yourself that you look wrong, and that your penis is wrong. Tell yourself that you are supposed to be curvier and rounder, and you're supposed to have breasts. Construct a mental image of what you would look like as a woman, and focus on that in contrast to what you see in the mirror. Find every variance, and focus on how it
should look. Repeat this a few times a day. When someone mentions your outfit or your looks or glances at your body, tell yourself that they're judging you for how wrong you look. Tell yourself they're not seeing the real you. Imagine how they would treat you if you looked like the woman that you're supposed to be, how they would interact with you differently. Make this a repeated mantra for one week.
Then come back next week and let me know how you feel about your actual body.