It probably depends on the hardiness of the breed - both fish and dog.
Seems like the carefully bred fancies have more delicate digestion. Ingest air when the food floats, get constipation and blockages and stuff. I've never had a problem with my comets, and they try to eat anything that falls in their barrel. Though they do spit out unsavory morsels. One was even making kissy noises the other day, pulling air in his mouth at a crease in the pond liner at the surface. And he wasn't full of air later. Strange fish!
Then I've known little Pomeranian and Shitzu that get diarrhea, constipation, or impacted anal glands if they eat anything slightly unusual. Then there's the Tijuana ranch dogs my friend had that would eat ANYTHING that fell on the ground and it wouldn't phase them a bit. One ate a cassette tape. When it came out the other end he had beads on a string. Scared him though - it was stop, grunt, run away, stop, grunt, run away. And that strange thing wouldn't stop chasing him - until he got to the end of the tangled mess. Now THAT was funny - and I don't think he was harmed aside from the psychological trauma.
Still, I don't think I would put cheese in with my fish for fear of the strange bacteria it may have.