Does any animal species have two sexes and more genders?
Many organisms have multiple sexes. For instance, here is one which apparently has 7 (and each sex, or "mating type" can only mate with the others). The "gotcha" is that, mostly, they're unicellular organisms where mating in itself is weird to begin with. I haven't ever heard of a metazoan which truly has more than one sex (there may be variation within the males or females, but I think the sperms or eggs are always interchangeable).

Gender, I think, is more of a social term relevant to human behavior. Especially nowadays, so much of the literature and understanding of what is referred to as "gender" is closely linked to things such as gender identity, sexuality, gender roles and personality that is difficult to see how you could meaningfully speak about the "gender" of (non-human) animals.

It's analogous to the idea of "animal language" - certainly some animals may be said to behave in stereotypical ways which influence their mating behavior. Does that mean that what is written about human gender applies? I suspect precious little of it would. I would be quite skeptical of someone discussing gender (in the modern sense) for even uncivilized humans, let alone non-human animals.