(NCed -- you probably know what my former, similar, MN name is; this one is here to stay for certain reasons.)
Rachel Rooney wrote a lovely book for children called “My body is me”.
That is basically how I feel about it. I don’t believe in a soul or inner essence. My body, all of it, is me. My brain is part of my body not separate from it, just like my lungs, heart, kidneys.
Speaking only for myself alone: I don't feel "my body is me". I feel that I HAVE a body, I AM not that body. I inhabit it. Yet still, I do not have an immutable "feminine essence": that body is the reality in which I must go through this life, and I have to accept that body with all its female qualities, the things it can or cannot do, they way it is accepted or not accepted by society.
It is a woman's body; it had periods and got pregnant, bore children, breastfed, went through menopause.And because it is a female body, I am a woman, and get to call myself woman, just like every other adult with a female body. Nobody else gets to do that.
You can't conjure yourself into womanhood. There is no way that someone who has not experienced what it is to live in a woman's body, can be a woman. The two, personality and body, are linked together, the one shaping the other and dictating to some extent its role in life, such as the ability to conceive and give birth.
And just to be quite clear, the OP is right about religion. Though I don't belong to any religion, I do feel a "higher, all-intelligent power" and believe that it is that feeling that propels people towards a particular religion. I understand it perfectly, as I feel it too; but because it is a feeling, there is no way I can or would persuade another person that it is the truth and it is the only way to be. It's a feeling that absolutely cannot be explained rationally. (I also know what it is to be an atheist, as I was raised to be one by very atheist parents!)
And it is quite unlike the personal sense of being a woman, which CAN be explained rationally, as I did above. My sense of the personality which lives inside my body is very much influenced, not only by the functions of my female body, but by the way society has treated it. Being harassed by boys as a teenager, being coerced into having sex which I really didn't want, being spoken over and interrupted by men with louder and more forceful voices: all these things have shaped my personality and made me me, the me that inhabits this female body. And while my personality can change, and has changed enormously over the past 69 years, the experiences of living in a female body is part of the process of making it what it is today. Not ONLY those female experiences, I might add!
But I am a personality within a body, and I disagree with the general feminist stance on MN, and that book, that "my body is me". I am more than my body, but I have to and do take my body seriously.
So, that's my own take on it. Sorry this was so long, and hope you can make sense of it.