But if it's not defined by outward characteristics (I'm with you on that) and it's not defined on biology, then what are we left with? Just this vague idea if 'feeling like a woman' which is a circular definition and doesn't include many, many people previously viewed as 'women'.
I've seen definitions/discussions of terms like cissexual and cisgender which talk about people whose experience of life is of their mental and physical sex/identity being aligned, and feeling as if certain gender pronouns typically associated with their biological sex correctly capture them, and so on.
So maybe that is the answer you are looking for. When you are in the "right" sexed body and/or being described with the "right" gender pronouns and/or other people are treating you as the "right" one of the two, then everything feels "normal" and as it should be. This is perhaps what it means to feel like a man/woman. Certain things feel "normal" and "right" to you.
Not the sort of thing most of us would notice in and of itself. You would either have to be trans or do some kind of Freaky Friday thought experiment where you get body swapped, to then have your feeling of being a man/woman made starker, since that feeling of normalcy and rightness will not be there if you are in the body of the "wrong" sex and/or are being described with the "wrong" pronouns and/or having everyone else acts as if you are the "wrong" one of the two and etc etc.
What I am suggesting in a nutshell is, feeling like a man = feeling as if a certain body and/or being referred to as he and/or being considered as a man, feels right and proper, and as if being in a a certain other body and/or being referred to as she and/or being considered as a woman, feels not right and not proper. With suitable adjustments made for feeling like a woman.
Well, it is not circular at least.