So you could remove a part and replace it and he'd remain the same Mr Potato Head, though not necessarily unaltered, any more than any of us remain the same over time
also, the pieces of Mr Potato Head were able to move around on their own, without the usage of a body, which is where Mr Cucumber Head and also Mr Tortilla Head happened
in Toy Story 2, Mr Potato Head had angry eyes, which he chose to swap out, so I believe that the consciousness can choose which parts it wants to activate as its own
shitpost answer: in his hat
there must have been some magic in that old plastic hat they found