The “magic law” is just the consequence of what it means to be the “same” person. To be the same person, you have to, among other similarities, take up the same spatial-temporal space. This is why if we ask “is Bruce Wayne the same person as Batman” one of the first thoughts is “you know, I’ve never seen them in the same room before.”
Maybe what you’re getting hung up on here is the split. Let’s imagine there is one river (river A) which goes for a bit before it forks and becomes river b and river c. In some sense, we could say that both river b and river c are river a. But if you’re river b, then river c is not the same as you, and vice versa.