It seems to me that once the copy is made of you, then immediately you and your copy's histories will begin to diverge. So, you and your copy clearly have distinct identities. However, it seems that both have an equal claim on the "pre-copy you" identity.
This is indeed an interesting arrangement and it may one day be of practical importance - the scenario described in Robin Hanson's The Age of Em involves large quantities of copies of a comparatively small number of biological people. Even if uploading can't be done in a manner that is non-destructive to the original, or even if the uploading process creates a less-than-exact copy of the original, it would seem like once a person is uploaded, it should be possible to make exact copies of the EM non-destructively.
You have two beings with the same history and same memories and same bodies and same claim to possessions/friends/legal status/partners walking around in the world? This has never happened in the history of the Earth!
Actually, identical twins are a special case of this - where the copy is made shortly after conception. (Albeit because the copy is made so early, there are no shared memories, possessions, etc.).
I had tried head on attack on the question before and surpisingly it results in extreme complexity of answers. http://lesswrong.com/lw/nuc/identity_map/
I think I should elaborate. To solve identity problem we should solve before several other problems:
Nature of reality including role of the observer in QM
Nature of consciousness and qualia, and other ideas connected with the question 'who is I"
Nature of human idea of identity which has social and biological basis
Correct decision theory which is able to deal with many copies situation
Looking at edge cases is pretty productive for mathematicians - you don't go around seeing things like the Cantor set or functions that are continuous everywhere but differentiable nowhere in everyday life, but weird things like those tend to be important for advancing understanding.
On the other hand, hard cases make bad law - accommodations for weird and egregious situations (like the teenager who can't legally consent to sex but can be held legally responsible for child support) tend to become loopholes that get exploited to make worse outcomes in more common situations.
Guy who wants to do philosophy with me: "If you made a perfect copy of yourself and destroyed the original, would that copy be you?"
Me: "Oh yes, of course. Trivially."
GWWTDPWM: "Ah, but what if you didn't destroy the original immediately, and then..."
Me: "What? You have two beings with the same history and same memories and same bodies and same claim to possessions/friends/legal status/partners walking around in the world? This has never happened in the history of the Earth! This is bizarre! Unique! Interesting! Let's plumb the philosophical mysteries of this novel arrangement, and what it means for identity!"