Ha! Nice try, but twins are born with separate identities. The OP suggests that at the split, both people would insist they were the original person, and both would be right in many ways..
Why on Earth would both insist they were the original?
More accurately, why would either insist they were the other, any more than birth-twins do?
One of them steps into a teleporter on Earth, and steps out again a moment later, deeply, existentially aware that he is not on Mars.
The other steps out a moment later, deeply, existentially aware that he
is on Mars. Everything else follows from that, and is simply a matter of nomenclature and bookkeeping.
Aside from the obvious absurdity of the one on Mars claiming he is the one on Earth, why would he want to, any more than any twin would want to claim the identity of their sibling?
Sure, twins may sometimes claim to be each other, as a prank, or as a crime, but these shenanigans are merely parlor tricks; they don't even begin to touch on the fundamental self-identity of either twin.
I mean, sure you could brainwash one or both of them, so that they no longer remembered which one of them was on Mars and which one of them was on Earth. And yes, that would raise some interesting questions about self-identity.
In fact, that's exactly what happened in the blockbuster movie
Total Recall, minus the cloning bit. You know how many people spend any time at all worrying about who was the
real Hauser? None. They all stop caring by the time they get from the movie theater back to their car. Because, from a practical standpoint, identity really isn't all that hard to sort out.