I think better formulations might be:

How can a body & an identity be "mismatched" (what has one got to do with the other)?

or

How can gender be both an immutable identity & an ever-changing social construct?

Male biology begets male socialisation, which begets male behavioural trends, even within the population who claim to have a cross-sex identity. I guess you could distinguish the sexed body & the resulting cultural conformity to the role associated with it as male vs. masculine, respectively, but most people think of masculinity & femininity as being limited to the superficial (the way you dress, walk & talk), even within feminism, or gender theory.

Judith Butler for instance quotes Simone de Beauvoir when talking about femininity. So at first it seems like she is using Beauvoir's understanding of femininity: the conditioning received by girls, but Butler suggests that anyone can be feminine (implying that even people who haven't been conditioned the way that girls have can be feminine), which seems to suggest she is using a colloquial definition of femininity, making her interpretation as hamfisted as Bruce Jenner's.

What is femininity? Is it limited to a glittery exterior? Is it female socialisation? Is it both? Is it even more than that? What if we extended the definition of femininity to include all of the roles women are expected to embody? In that case this new stipulated definition of femininity would also refer to things like sexual objectification & the nurturing of children, the elderly, partners, each other & anyone else who happens to be passing by – two huge features closely associated with women, but almost never associated with femininity

So if you reduced femininity down to its root (using the word femininity to refer to how an androcentric society views womanhood) it would come down to five types of objectification for male use:

1. a decorative status symbol for men
2. a fetishised masturbation aid for men
3. a de-clawed, domesticated, infantilised pet for men
4. an indentured servant for men
5. a vessel for men to perpetuate themselves

If that's the social construct, can it be an identity? Notice how female biology is also interwoven into femininity. If someone is inculcated with these ideas from birth, the same way men expect the above from women is the same way women expect it from themselves & each other. Every word is culturally contaminated, including woman. Within a scientific context woman is just a female adult within the human race, but dictionaries don't include connotation. If society is androcentric/patriarchal & therefore systemically sexist, then the people within it are at least implicitly sexist, which would include their view on womanhood. In this way being a normative woman & being normatively feminine are exactly the same thing.

We are all aware of sexist slurs, so there is no need for me to repeat them for clarification, but in a sexist society these are not slurs, but synonyms for 'woman'. 'Woman' is just a formal way of saying all those slurs. Some girls try to opt out of this by calling themselves non-binary & when that doesn't work they call themselves men. The word 'woman' is not the source of the sexist contamination, but as long as it is contaminated, it will continue to contribute to sexist ideas. Women have sexist notions of what being a woman means (which I have previously referred to collectively as femininity), so identifying themselves as a woman is like a positive feedback circuit of these sexist notions of femininity, influencing their identity.

The obligatory "but what about exceptions?"

Obviously almost no woman will perfectly conform to all five ways of objectification, but there is certainly a pattern of behaviour within the female population that conforms to femininity. So the questions are: are behavioural trends within a demographic the result of a shared social identity? And is identity the result of socialisation? If the answer is yes to both, then "woman" can be an identity.

DSDs:

The only way woman could be a cross-sex identity is if male biology is mistaken for female biology "from birth", by everyone including the child itself. Then the domino effect I outlined in my opening paragraph would go in the other direction. This is the case for women with complete androgen insensitivity syndrome. It could be argued that their biology can be said to be more female than male, but that would be ignoring the fact that nature produces organisms that can reproduce asexually. This "default setting" would look a lot more female to our eyes than male (even though it cannot be said to be female, confusingly, science usually refers to it as female nonetheless), bc if a human could asexually reproduce it would require primary & secondary sex characteristics that currently only exist in the female population.