Actually not exactly the same. The posessive pronoun ("our" etc.), like the preposition "of", can mean many different relations, not only ownership. In the case of fish it's hard to imagine other possibilities, but here are two:
A fairy tale where fish can talk and have family relations. Father fish and mother fish say to uncle octopus about their child "the fish is ours".
A cooking competition! The referres, who have tasted the dishes without knowing who made which, have just announced "the winning dishes are the fish and the caulliflower. Will the proud cooks step forward?". Two excited girls step forward, and declare "the fish is ours!"
In both cases, you wouldn't use "belong". So I agree that where you say "the fish belongs to us" you can always* say "the fish is ours", but belongs is more specific. Same in Hebrew.
- Actually... this is far fetched, but imagine in this cooking contest, the two girls say angrily "the fish is not just ours, it belongs to us! We didn't intend to give it for the competion at all!"
So when do you use שייך versus שייכת? The only difference I see is שייכת was used with a name or one person specifically. שייך is used without a name and references belong to multiple people, to us? Can someone please clarify usage and perhaps also add it to the word description in the course because this is very confusing. Is the difference in plural or singular?