As a new piano player I find the entire concept of doing two separate things at once massively challenging. Even when both parts are incredibly simple and I can sight-read both lines mentally easily, I start playing and my hands just "lock up". Even playing a simple 3 note repeating pattern (like a chord triad 1-2-3-1-2-3) with my left hand, the moment I try and play any different rhythm with my right, it all falls apart. As someone who is used to learning stuff easily, this is a strange concept. I can almost stand outside my own brain and watch all the gears grind to a halt at the simplest thing.

This isn't a question about how to get better at playing but about how it works. Does my brain really learn how to do two things at once or as I learn piano, does my brain understand how both hands are not doing separate things, but are actually working together? e.g. does a good pianist look at a complex piece of music when sight-reading and see two separate lines, or one thing which happens to be shared between their hands?