[masklinn]

From other comments, you would probably describe the pet’s features (likely with a fair amount of confusion / fuzziness), or some feature-descriptions might be stand-ins for specific animals.

For instance imagine that rather than a cat your pet is a chuckwalla, while that name is quite precise, it is almost certainly useless to the average conversation partner because most people have no idea whatsoever what a chuckwalla is (or that there’s something called a chuckwalla in the first place).

[Kwpolska]

If you have an unusual pet, the name is useless, agreed. But everyone knows what an average pet, like a "cat" or a "dog" or a "guinea pig", looks like, and it would be weird to describe what a cat is in response to this question in most non-conlang languages.