I don't think they would without the trans doctrine use that wording and mean it literally -- the whole "I feel like a woman" thing. It's a carefully crafted language construction that confuses "is" with what allegedly "ought to be". Essentially fake it until you make it. If you say it enough times, it will become true. They want to be women while being aware they're not, but since that's physically impossible they'll settle for the next best thing of essentially engaging into roleplay of one, taking into account the quality of this emotional experience depends on participation of outsiders as well. Now as for why they feel the need to do that, there are probably a couple of related as well as distinct reasons, the main one being that their personality as it is is more favorably seen through the lense of femininity rather than masculinity.

Basically, society attaches different value judgements on the same things depending on who is doing them. This is very obvious given the existence of many gender norms -- there are behaviors that you get rewarded for if you're female and that you get punished for if you're male. As one example, engaging into rituals that suggest you sexually objectify yourself or are trying to induce other people into sexually objectifying you is only considered acceptable (and even necessary) if you're female. A woman doing this "right" will perceive some sort of (toxic but still possibly reading positive) acknowledgement for it. A male will receive derision and mockery.

So to them it's not as simple as telling them "you can do everything women do", yes, it's physically possible for a man to put on makeup and leave the house as such, but the social penalties of it while you're male (or even simply a lack of social approval) is still present. They don't only want to perform femininity, they want to perform it and be applauded for it. They want to be perceived in such a way that the value judgement of their behavior ends up positive, and they see being seen "as women" as the obvious solution to it.

That's at the essence of it, the entire "this is my identity" and "I literally know I'm a woman" are just cliches meant to translate to their environment just how unwavering they are in this desire. They don't believe it in the literal sense themselves, it's a thing of telling people what they need to hear in order to manipulate them into proper behavior, that is treating MtTs "as the women that they really are". Though of course, with this becoming the party line probably more and more of them do end up believing this literally, they internalize this is - ought to be confusion and think their desire to be treated as female is some evidence of innate femaleness in their brain. But that again just makes it into a semantic argument, because other females aren't defined by this mysterious brain quality -- their problem is precisely that as much as they twist themselves into pretzels they know they're distinct from females, it just depresses them or pisses them off.