I'm only myself when I'm alone, my personality is different for different people.
You can have a core personality and have fake personalities for other people. They're just different 'masks' to blend in to different social situations. I think everyone uses a mask to some degree.
For example: I might pretend to care and display sympathy when someone is telling me something they consider a sad or tragic event, but underneath the fake caring, I really don't give a shit.
I think my core is very primal and instinctual. At my core I'm just a predatory animal that is selfish and self-serving and my core personality reflects that.
"You can have a core personality and have fake personalities for other people."
Not in my case, because who I am around people is entirely about my audience, every person is a unique snowflake, hence my personality is never uniquely the same to anyone, in contrast, I think empaths generally only have one personality.
Your case sounds almost exactly the same as mine except I get the impression that you don't fake emotion when socializing. You essentially become whoever you need to be at the time. That's basically how it is for me, except that I need to fake certain emotions when necessary because I don't actually feel whatever is expected of me at the time. Those fake emotions mask when I'm truly feeling, which is usually apathy.
I read somewhere in a post that you have a daughter. Do you care about her? Because it is instinctual for a mother to care for her young.. Idk, I'm just curious I guess. If you do, I wouldn't be surprised at all. Please don't be offended if I asked in an inconsiderate manner