Science is mostly based on materialism — we are the configuration of our bodies, so an identical copy isn't "just" a copy, it's you, again. This doesn't feel like it meshes well with our subjective sense of continuity. If I should suddenly die, it seems to me there's all the difference in the world between the me writing this resurrecting and a copy of me.
As far as I know there's no real answer to this. "People have souls that are made of energy" is often used as a placeholder, but it just moves the problem. What if you copy the soul, etc?
I can offer just one point out of recent science that might be useful for your story, especially if you're willing to bend physics a bit. You may know that quantum physics implies a certain amount of fundamental uncertainty in the position and momentum of particles. I cannot take particle A and copy its state exactly to particle B because I cannot measure the state of particle A exactly. This is a universal law - if I could copy A, then I could simultaneously measure both copies and get more information about the particle than can exist.
The one exception to the no-copy rule that has been found is that if you have two entangled particles (call them X and Y), you can take a combined measurement of A and X, then apply that measurement to a combination of Y and B in such a way that particle B takes on the exact state of A. The result of the operation ruins the entanglement of X and Y, and also changes the state of A. Thus you can "teleport" the information from the location of X to Y, but are fundamentally prohibited from copying it.
Does this mean you "can't copy identity?" Not necessarily — we're changing all the time anyway, and just because you can't copy a given particle doesn't mean you can't copy something much more macroscopic - say a computer program. Does consciousness exist at the level of a computer program, or deeper in uncertainty?
For the sake of your story, you can easily say the latter. Add some hand-wavey bits about establishing entanglement over great distances and you have a can-transport-but-not-copy device with room for singleton identity (and also some interesting possibilities to imagine on what you get if you do try to copy - what do you have if you have all the neurons in the right places macroscopically but the quantum states are all randomized at a deeper level? A new person? A gibbering deranged monster? Does the scrambled brain fundamentally just not work?)
