Gender identity goes much deeper than just gender roles. Nearly all trans people experience gender dysphoria, which is usually split into body and social dysphoria. Body dysphoria refers to distress over your body not feeling right. E.g. I feel disgust and unhappyness when I see myself in the mirror and notice masculine features whereas seeing feminine features makes me very happy, and makes me feel good about myself. The idea that I might not ever look like a normal girl is enough to bring me to tears. I feel strongly like my body Should be female, and that its male features are Bad and Wrong and shouldn't be there. I often get phantom body parts; feeling like I have boobs or a vagina and if I'm a bit tired it can be confusing for a moment to realise that they aren't actually there. Social dysphoria refers to the same sorts of unhappiness and distress but caused by being perceived as male. Being called a guy sucks, being called a girl makes me happy :). I feel deeply uncomfortable wearing masculine clothing, and desperately want to express my femininity, be treated as a girl and interact socially as one. The distress from these is bad enough that 41% of trans people attempt suicide by the time they are 20. Not all trans people have both body and social dysphoria.
Unfortunately, as it turns out otherkin are a real thing. Many of them interpret it spiritually, which I disagree with, but that's why it is usually interpreted as non-human souls or reincarnation or whatever. But a large minority of otherkin (including myself) experience species dysphoria, so called because for most otherkin it feels identical to gender dysphoria (30% of otherkin are trans/nonbinary/intersex, so we have a pretty broad basis for comparison. This also suggests the conditions are related). I get body and social dysphoria over my species -- the fact that I don't have fur or claws really gets to me, and can make me feel physically uncomfortable and like I am naked even when fully dressed. I have phantom wings and a phantom tail basically all the time. I can't say that I am human without feeling like I am lying, and I have a bunch of animal-like mannerisms that I wish I could express publicly (this is less of an issue for me, as they are mostly related to showing affection and my boyfriend is very understanding). And species dysphoria can't be treated. There are things I can do to feel and look a little more like myself, and I have heard psychotherapy is helpful for helping you cope with the pain, but for the most part I'm just stuck like this. It doesn't help that my species is part of my fundamental identity, so I would absolutely choose suffering for the rest of my life over having my identity changed, just as I would choose being a depressed and dysphoric girl over being a happy and functional guy. It doesn't matter how much it hurts, it is me.
It should probably go without saying that you shouldn't trust tumblr on this stuff, since it is basically liberal 4chan.