Uh, no, it's completely unique. What does variety in a person's biology have to do with whether his particular variety is unique from that of another person? Nothing at all. I haven't seen a calculation recently but how much do you want to bet that the odds of someone being born with identical genetic structure to another person are more astronomical than the amount of atoms in the universe?
What are the odds of identical twins, again? Not astronomical. Also, it was you who first brought up clones. Human clones will happen at some point (if not already).
I never said it should be the whole definition, but the argument being made is that it is totally irrelevant to the identity of the person. This is the point that I think is absurd. The burden is on your side to argue that DNA has no bearing at all on the identity of a person. If it has any bearing at all then it can be added to a list of evidence that an entity is a unique person.
Let's go back to your original statement:
"I think it's a fairly common-sense thing to say that one's body and organs are the result of having a certain set of DNA, and that beings with different DNA are by definition not the same person."
You are correct that DNA largely determines the structure of bodies. But you are making a subtle claim by talking about "beings with different DNA", begging the question of how you identify separate "beings". While you can say that two beings with different DNA are not the same being, it's circular logic, because you've begun by saying the two beings are separate. And having "different DNA" is not what defines a person, as proved by the multiple counterexamples you've been presented with.
"the argument being made is that it is totally irrelevant to the identity of the person"
No, the argument being made is that it's not sufficient to identify a person. I don't need to show that different DNA has no bearing on personal identity, since I'm not making a positive claim. I'm just agreeing with some others that having unique DNA is not actually a key characteristic of personhood, and hosting tissue with different DNA doesn't mean you are hosting a different person.
I do actually agree with you that gestating inside of a woman's body is not sufficient to define a fetus as "not a person". I think defining a person is an enormous challenge, so I'm not going to try to do it here, but I really don't think checking for unique DNA helps us decide whether something is a person, and you haven't shown in any way that it would help.
Denying the antecedent fallacy. From the fact that unique DNA does designate a unique person (assuming it does) it does not follow from this that a lack of unique DNA does not designate a unique person.
Unique DNA can exist in a mutated cancer cell. It doesn't designate a person.