It's not a misunderstanding at all. In order to know that you feel the same way as someone else feels, then both parties need to be able to describe the same experience in similar ways..
Your stuck in the same loop.
" In order to know that you feel the same way as someone else feels, then...."
you don't need to know what anyone else feels that's the entire point.
Cognition doesn't work like that - it's 'thinking' that takes place in your own brain, it doesn't depend on knowing someone else's internal experience.
A chromosome is a chromosome. Your knowledge of your chromosomes is a cognition generated in your brain. It can develop through social learning—you know what a chromosome is because someone explained it to you—but having that cognition doesn’t depend on knowing what someone else’s internal experience of thinking about their chromosomes is like.
Gender incongruence is a neurocognitive/psychological condition whereby a person’s internal recognition or representation of themselves as male or female does not correspond with their physical anatomy.
The person is not deluded about what anatomy they have, but despite this anatomy, their internal perception/ recognition of self remains persistently at odds with those physical characteristics.
This is not something that a person can typically control, and it doesn't require knowing how other people experience being male or female.
Now you might say that this cognition has developed because the person has learned that girls like dresses, and they like dresses. But that is a different proposition from a claim that it requires insight into how someone else experiences being a girl. It doesn't.
And that explanation - the role of stereotypes - cannot simply be assumed because it seems intuitive to you. It does not account for trans girls who don’t have stereotypically feminine likes, preferences or interests. Having stereotypically feminine interests is not the same thing as recognising oneself as a girl. A trans girl can understand perfectly well that boys can wear dresses without that resolving her incongruence. She might not even like dresses.
And if you listen properly to accounts of gender incongruence, a recurring theme is not “I’m a girl because I like playing with dolls”, but a direct recognition of self. A conversation like the one I had with my 4 year old earlier this evening is much more typical-
"how do you know you are girl?"
"I just know it".
"because I am"
"I feel it in my body"
"I see it".
If you tell a gender incongruent child - "you can be a boy and play with dolls, where dresses, be whomever you want to be!", she is likely to reply:
"I know I can. But I'm a girl".
There is also research relevant to claims that gender identity is simply a matter of stereotypical interests or being socially recognised as a boy or girl.
Gülgöz et al. (2019) measured children’s identification as boys or girls separately from their preferences for toys, clothes and playmates. They found similar variation among both trans and cis children: some girls had strongly stereotypically feminine preferences, while others did not. Being a girl, or identifying as a girl, and liking stereotypically feminine things were not interchangeable.
Olson and Enright (2018) found that trans children endorsed fewer stereotypes and were more accepting of gender non-conformity than gender typical comparison children.
Rae et al. (2019) assessed gender-nonconforming children before some later socially transitioned. Those who subsequently transitioned had already expressed stronger identification with the other gender before changing names or pronouns. Their identification therefore preceded their social transition.
None of these studies settles how gender identity develops ofc. But they challenge the simple assumptions that a child identifies as a girl because she likes “girls’ things”, or because other people started calling her a girl.