Right. You'd have a memory of cloning your mind, and the copy wouldn't have a memory of that. From the first boot of the copy, it would be operating with a different set of data than you. Also the fact you exist along side of it, means it's not you. It's a copy of you.

As long as we're dreaming up stuff, why do we assume the consciousness wouldn't be shared? what if you woke up and there was one mind but two bodies?

For myself at least, consciousness is the product of the body and in particular the brain. Without any kind of high bandwidth connection between two bodies, I don't see how your proposal could work.

That said, I wonder if that offers a possible way to 'transfer consciousness' between a human and their robot copy - link both brains together and then once synchronised (whatever that means), shutdown the biological half. Like a sort of extended Ship of Theseus idea.

That's an interesting idea. I think as long as the continuity of consciousness happens then it doesn't matter and you would negate the copy problem.