FYI: strictly speaking, кошка is a feminine word, for a male cat we prefer кот. But as biological species (felidae), all the cats are called кошачьи (from кошка).
As for the word собака, it usually used for both genders, but for masculine we can also use пёс, from which originated a name for the biological family псовые (canidae).
Out of curiosity, does this sentence mean "Do you have a cat a cat or a dog?" As in if the person is asking if you have one or the other (implying you must have one of them), or if you have either (with no implication of owning either animal) or can it be used to ask both? Probably a dumb question, but I hope that makes sense.
That's not a dumb question at all. It's a good question. I know around four languages where the "or" in everyday life means... "any combination of the... things" (uhh... that is "either or" I guess) and the difference is only noted when the topic is (mathematical) logic really. It would be interesting if it was different in Russian.
Is 'или' inclusive ("and/or") or exclusive (only "or")?