There is actually no equivalent for pet in Italian, they're indicated by combination of words as "animale domestico", "animale da compagnia", and similars; but those names are used only when necessary, most often one just says animal(s) and you can guess by the context.
That sentence is a bit weird...does any other translation than "pet" make sense here? Usually you'd either say "I've got a [insert animal here]" and it would mean "as a pet". To say something like "I've got mice" generally translates as "Yuck! Got vermin in the house, it's mice" Never heard anyone say something like "I've got an animal. It's a tomcat" O.o