So I know what you're talking about with piano. I'm definitely not a pianist, but I've taught myself (and promptly forgotten how) to play a couple of simple two-handed pieces, and I have the same difficulty you describe in controlling my hands independently.

But at least for me it's not that each hand wants to copy the other. I can play the left-hand part by itself just fine with the right hand still, and vice versa. The problem is that I have a hard time having each hand perform its task simultaneously with the other.

I play wind instruments more than piano, and I never had a problem (and never heard of any other musician having a problem) with the right and left hands doing different things because they are synchronized. You arrange both hands in a certain way to form a single note, and then articulate that note. Both hands are doing different things, but they are doing them together for a single purpose. It's giving each hand a separate independent task that causes problems, and the wires get crossed where both hands want to do one of the tasks.

Another issue might be similarity between what each hand is doing. While playing computer games, I have no problem controlling it with one hand on the mouse and the other on the keyboard, issuing separate commands through each device. But when I tried attaching two mice to the same computer and controlling two cursors, one with each hand? Forget about it.