Imagine you fall in coma and sleep long enough that all your body cells have been changed. Do you wake up a different person? I don't think so.
What you perceive as yourself is the experience of being aware of everything you have learned in the world. An identical brain, whether created by teleport or running in simulation, is you just as much.
So yes, both simulations will be sure they are you (and in a sense, both will be you—if we allow the simulations, we must abandon the idea of there being only one “you” at the moment). The funny thing, these simulations will immediately diverge by obtaining different experiences, and thus can be thought of “forks” of you.