Copying this from a comment I made on Facebook in order to have it easily accessible for future noodling. This is of course only how I experience dysphoria, not how anyone else does. The original discussion was, in part, about the extent to which dysphoria would still exist in the absence of gender stereotyping. I have made some minor edits for clarity.
I'm not "in the wrong body"; this body is very much part of me, and there are things I really like about it. But there are parts that feel stunted, like they never fully developed like they were "meant" to, and at times they ache as if they were still desperately trying to. Descriptions of phantom limbs from amputees often resonate with me. I assume this would not go away in a perfect society. It has got worse with perimenopause and seems to show some cyclical variation, so I imagine there's a hormonal factor involved.
Also, some of the parts I really like are the very ones that cause people to guess my gender wrong, and that causes a real psychological tension. I have to choose every day whether I want the cognitive dissonance of hiding those parts as if I were ashamed of them, or the cognitive dissonance of leaving them visible and being misgendered. This part would clearly improve if societal etiquette changed so that it was understood to be rude to guess someone's gender without being told, or at least rude to voice the guess.
Then there's a gender role/performative component, which expresses itself as a feeling that I'm constantly failing at "being a girl" by looking wrong, acting wrong, thinking wrong and just generally Being Wrong. This part has improved considerably since I gave myself permission to stop trying to be one, but there's a residue that would probably require societal permission to get rid of. For me, full societal recognition of nonbinary gender(s) would probably do it, but total abolition of the gender binary would also work for me (and for agender people? but maybe not for strongly binary-identified people?) Even then, there may be a biochemical component that would not disappear, because I feel the looking wrong part is linked to my bulimia, and that gets worse with certain nutritional deficiencies and could presumably still happen without sexism. Society not being so damn fat-shaming would surely help, though.
Summary: In a perfect society I'd probably still have body dysphoria and maybe a small amount of psychological discomfort. I might still define as trans because although we wouldn't be assigning gender at birth any more, I might still have self-assigned as a girl before the hormones kicked in enough for the body dysphoria to become noticeable. But I'd have much less cognitive dissonance and everyday life would be much more comfortable, so none of these issues would be the grave threats to mental health that they are now.