Not as an absolute way of determining rights (if it was, it would be justification for racism), but seeing it this way helps somewhat. For example, one could ask 'why should I respect a cat more than a bug?' and the answer would be 'the cat is genetically more similar to you, so it is your nature to respect it more'.

Are you kidding? Preserving creatures that are more similar to yourself is how evolution works. What if, for instance, all of the human race was wiped off the face of the Earth. Would you, biologically speaking, prefer the Earth to be repopulated by apes, or by rats? The obvious answer would be apes, because you, as a human, want the future to have the closest thing to yourself in it. Objectively, it's just as 'good' for the Earth to be populated by rats, lizards, fish, bacteria... or even nothing at all. But living things don't see the world objectively. The very thing that makes a thing living is its subjective desire to preserve that which is closest to itself.

It's true that we are getting somewhat off topic here, but the original question was 'if some animals get to have rights, how do we determine which?' There is no objective answer to this question! If you claim that all animals should be equally respected, then we can't live our lives at all: even collecting fruits and vegetables involves the suffering of some animals.