I'm with you. I don't see any reason to suppose that some future person who happened to share your DNA would be you, no matter how similar the course of their life might be to yours. And that's even before we get to the question whether time in infinite (quite possibly not) and whether, if it is, it has the implications that this argument supposes.
It isn't entirely obvious that I, say, could have been female from conception, and the assumption that the fertilizing sperm was different certainly leaves that possibility open. But if I could have been female, then your great-great-great grandfather (let's say) could have been female, and one of your great-great grandwhatever's parents would have had a hard time conceiving a child together.
Maybe that isn't the sort of possibility you had in mind. But it's not obvious how to restrict it and still get plausible results. Let's suppose you could have been the result of fertilization by a different sperm. What's so special about the sperm? Why not a different ovum, too? But now consider that other ovum and sperm. The latter could have fertilized the former even if the ovum and sperm from which you were actually formed still got together. But then are you your own twin? I don't think so. So it doesn't look as if you could have been the product of a different ovum and a different sperm. But if not, then why should only one of them be required? It's not obvious how to answer this question.
I can well imagine that there could be such evidence. But for the evidence to be truly trustworthy, it would have to be collected by people who were neutral, more or less, on what it was supposed to demonstrate, and the evidence would have to be in some sense replicable, and to stand up to critical scrutiny by reasonably neutral parties. So far as I'm aware, there is no evidence for reincarnation that meets anything like this sort of standard, however.
Try this question: Could you have been your sister and your sister been you and everything else been pretty much as it is? I find it kind of hard to get my mind around that: In what precisely would it consist that you were her and she were you? There are certain conceptions of the soul that would make sense of that: Your soul would occupy her body and hers would occupy yours. But even those philosophers attracted to a notion of soul have usually thought the soul was more intimately connected to the body than that: If we accept that kind of possibility, who's to say souls aren't switching bodies every time someone falls asleep?
So suppose we agree that isn't possible. Now it clearly is possible that she should have existed without you. But could it have happened that you should have existed without her but, so to speak, as her? What on earth is that supposed to mean? Either she exists or she doesn't, and if she doesn't exist, then you can't be her. (Perhaps you could have looked like her and acted like her and so forth, but that isn't what's at issue.)
But we can try to imagine that possibility a different way: Could you both have existed but been the same person? (This is Peter Lipton's question.) That one's not so clear to me. A case that might seem plausible would be that of identical twins. So here are Don and Dan, separate persons only because the blastocyst from which they were formed split early in its development. Suppose it hadn't split. What, then, should we say about Dan and Don? Would they have existed? It seems difficult to suppose that only one of them would have existed: Which one? But maybe it also seems difficult to suppose that neither of them would have existed, and, if so, then perhaps one should conclude that both of them would have existed but been the same person. But of course that's a special case, and one that probably doesn't apply to you and your sister.