You think so, too, does that mean we are the same person? I don't think so. The same holds for the copy.
We are not the same person, because we are different in some other characteristics and in our circumstances.
What's the reason you and your copy are not the same?
My copy and I are different persons because my copy has its own mind in its own brain in its own body. Just like you and I. Why would there be any more reason to think my copy and I are the same self, than to think you and I are the same self?
Again, how do you resolve making several copies?
[edited some formulations to make it clearer]