The argument about a copy being "identical to me but it's not me" is a very natural and intuitive reaction, but it's probably wrong. A very illuminating explanation is http://lesswrong.com/lw/pm/identity_isnt_in_specific_atoms and related posts.
I'll try a personal approach: imagine a country with no computers, no photocopiers, no presses, and a very strict law that forbids making copies of books which is indoctrinated from birth. There is only one copy of each literary creation, guarded in the National Library.
We can imagine that the people of that country have a very hard time to distinguish between "The Lord of the Rings" in an abstract sense and material object #1434 stored in the second floor, section Fantasy. For them LOTR is just that set of paper and ink. Even if someone secretly made a copy, letter by letter, it will be "just a copy", not the "true" LOTR. That's quite obvious to them.
And in a sense that's a bit our current situation with people (not exactly, because books are static and people aren't). We can't scan people-as-information out of people-as-protein-body and store them as people-as-silicon-body, so we mix the concepts. But it's just confusion due to technological limitations.
So in a sense we DO have souls, but it's a mathematical or information-theoretical soul, not a metaphysical one. It's always bound to a material instantation, but that concrete physical manifestation is not the same as the dynamic, interactive information process a person is.
So if technology advances enough we'll be able to copy a person, and it'll make no sense to ask which of them is really "you". Both are.