Does your name matter? (Case 2)

So, if your name changes, and your story stays the same; or your name stays The same, but your story changes; or your name changes and your story changes but you still have your body and mind, are you still you?
Here is a adaptation of the Harry Potter story so you can picture what I mean:
Are you unique because of your qualities that differ you from other humans or are you different because of your knowledges and opinions?