Both of you would be you, and you two would function separately, occupy separate spaces, and diverge slightly in ways that would only rarely make a difference to your personality.
But that's not the vertiginous question, which is "why am I me". I've wondered that before. However, it is nonsense. Naturally a person is that person, not some other person (and a tree is a tree, not some other tree). There's nothing strange about this. Why would it be otherwise? So the urge to ask the question really reveals some deep-seated misconception, or some other question that actually makes sense, and I wonder what that is.