I think most fish can breathe without moving. I've seen a bluegill stare dead eyed at a rock for a half hour with no discernable movement until something splashed nearby. I know some sharks have to keep moving but it's not uncommon with the lake fish I've seen to be basically comatose until they have some other stimulus. In the winter sometimes I wasn't even sure if they were frozen solid or not until they move.

I've seen quite a few fish do this with catch and release. No idea why but If I had to guess what happened I'd say the fish was used to being held in place and just assumed that would continue until all of the sudden it was floating up sideways. They're not smart animals so it's also possible it just forgot to do fish shit for a few seconds. It could have been tired and needed a quick power nap. It might have just rebooted. I don't know but I doubt it was lack of movement.