The issue that I see is that DNA isn't necessarily unique. For example, identical twins and species that were born asexually can have identical DNA. I don't really know any biology, so maybe there are other considerations too?
~~Also, hash collisions would happen randomly sometimes, so even if every individual had unique DNA, you couldn't use them as a unique identifier since there would sometimes be collisions.~~
Hope that helps.
Edit: I was wrong about hash collisions. Also, removed last paragraph because I was repeating myself.
Huh. That's way less likely than I assumed!
I think I was getting confused by how frequent collisions are in a hash-table. But those aren't actually hash collisions, they're hash-table entry collisions. So it makes sense that the likelihood of collisions is orders of magnitude smaller than I was thinking - the number of entries in any given hash-table is unfathomably tiny compared to the number of possible permutations that a reasonably-sized hash can have.