we have = nous avons
nous possédons = we own
In the sentence here, the story is not that we have/eat fish, but that we own/possess fish(es), in an aquarium or in our fish farm.
I understand your comment. But maybe both should be accepted? The hover-over translation did offer "have" as a possible meaning...
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.
I have never heard an English speaking person say they OWN some fish. We HAVE some fish?
Mine also rejected. Can someone explain how do I understand when to use 'some fish' and when to use 'fishes'?
Some fish or just fish should work here. Fishes isn't usually used in English, as fish is the plural of fish.