Do you know the man who is fishing? 낚시하는 남자 - the man who is fishing/the man who fishes. The fishing man makes no sense.

"Do you know the man fishing?" sounds more natural than the translation given

Fishing before man qualifies it. He's the sort of man who goes fishing, probably what we'd loosely call a fisherman. Fishing after man modifies it. He's a man who's doing something to do with fishing. Surely this should be, "Do you know who the man fishing is?" Fisherman seems to be 어부, fishing man 낚시꾼 at least in the dictionaries.

Do you know the fishing man