You’ve asked a number of different questions here.
1. “How does it feel to be a woman?”
We’ve already covered this, and agreed: there is no universal subjective sensation or objective metric for “what it feels like to be a woman”. Women do not all have the same personality, interests, emotional life or relationship to their bodies.
The awareness of being a woman is a neurocognitive experience generated in the brain, not a collection of feminine or masculine stereotypes. For most of us this awareness corresponds so unremarkably with our anatomy/ birth sex that we don't even notice it as feeling like anything at all. But , nonetheless, this awareness doesn't derive passively from our genitals, it is produced actively in the brain.
2. “How do I know I’m a woman other than by having a female body?”
That is a different - much more difficult - question- the "how" this happens . We do not yet know the precise neurodevelopmental mechanism by which gender incongruence arises. But there is nothing scientifically unusual about that. We don’t require a complete mechanistic explanation before accepting that a phenomenon is real.
We still do not have complete causal accounts of all kinds of things - like autism, sexual orientation, migraine, chronic pain, synaesthesia or Tourette syndrome. In fact this is true of almost all neurocognitive/ psychological phenomena, because our mechanistic understanding of how the brain works is still so limited. In each case we can study the phenomenon, identify patterns and biological or neurological correlates, and develop increasingly sophisticated explanations without being able to point to one simple mechanism and say “there it is”.
Gender incongruence is likewise a repeatedly observed phenomenon across cultures and populations: some people with male anatomy consistently experience or recognise themselves as female, and some people with female anatomy recognise themselves as male. The explanation is likely to involve some combination of biological, developmental and neurocognitive factors, and the science is still developing.
3. “How could a man know what it feels like to be a woman?”
They couldn’t know what another person feels like, any more than I can directly know what you experience internally. But that isn’t actually what a trans person is claiming. They aren’t saying, “I have compared my internal experience with every woman’s and discovered it is identical.” They are describing their own persistent experience of themselves and their relationship to their sexed body.
No stereotypes or circular reasoning are necessary here. The fact that we cannot yet give a complete mechanistic account of why gender incongruence occurs does not make the phenomenon itself circular, imaginary or meaningless.