Nothing about "we own" changes the "story". It could just as easily be a bag of frozen fish they possess for a cajun cookout. Someone with pet aquarium fish is more likely to say "I have fish". Have obviously meaning "own" here - because English speakers shy away from "owning" pets in general conversation. Livestock and fungible commodities you own.
In your examples, Sitesurf, at least in American English, we would tend to say "What kind of fish to you have in your Aquarium?" "I own a catfish farm, I have lots of catfish." This would be understood as "own," but as others have said, "We own fish" is not something we would say.