neither...nor with articles

Solution 1:

As @Edwin Ashworth says, we need some context. This is always needed because English is a context-dependent language.

Neither a P nor a Q

John: Do you have any pets?

Mary: I have one pet.

John: Let me guess. I bet it's a dog or a cat.

Mary: No it's neither a cat nor a dog. (This refers to a specific individual)

John: Is it a rabbit?

Mary: Yes!

Neither P nor Q

Mary: My goodness! What kind of animal is that?

John: I don't know. It looks weird.

Mary: All I can say is it's neither cat nor dog. (This refers to the species)

John: I think it is some kind of predator whatever it is.