I'm not worried so much about the uniqueness though, as the continuity. Well, no. It's both.
You cannot copy something fully, only partially, most of the time. You can copy most of the information without destroying the original, but not all. If some process exists such that multiple copies can be made, it's quite easy to say it's not the original item.
But if all of the information is copied, where the original is necessarily destroyed, then it's tough to deny that the copy is the original.
Consider the example where I want to teleport a book across the room. If I pick up the book and carry it there, we call it the same book. If I leaf through the book, uploading it into a scanner and then print out the book on the other side of the room ... then clearly it's not the same book. It's just a copy.