Me- i think people relate themselves to different animal a bit too much.
A fish can take a flop or two and be just fine- it is US humans that are much more fragile than the rest of species. It may hurt us or our feelings if that was done to us, but not the fish.
So long as the fish is not out of the water too long, has recovered from the fight, than getting tossed in is the best ride of its life.
If a fish needs its picture taken with two hands, thats how it is released- if it requires just one, it is dropped into the drink from standing up.
Come to think of it, it is usually done with a somersault action as well performing 180 degree flip head first into water.