For me, it's the physical body - biologically, what we are born as.
I get the feeling that when trans people say they "feel like they're a woman", or they "feel like a man", and their biological sex is different, that what they're talking about is their essence or soul. The problem with this is that not everyone believes in the concept of a soul, and even if they do, they don't necessarily believe a soul has a gender, and rather the soul is genderless and it just inhabits a physical body which happens to be one or the other during the lifetime on Earth.
Things get further complicated when you take into account that those who do believe in souls believe that each person is made as God intended, etc (therefore to mess around with what God intended is an atrocious thing to do).
We can go on and on... But you see where this leads... It will never be possible for the world to unite on this topic, for this very reason in my opinion.
We can not and should not force other people to have the same faith beliefs as us, it'll never fly. There are too many of us with too wide a variety of life experiences which leads me onto my next point...
Our own beliefs on soul/science depends on where we are born, when, education, the place we are born, grow up and live, the family we are born into and the people around us as we traverse through our time on this planet. And of course, what we see on the Internet and the good ol' algorithms (Netflix's Social Dilemma, anyone!?). So we also have to consider the element of social contagion, in my opinion, also here.
It's all very messy.