No it's not what they mean, Master Richard. The term refers to a specific style of angling, with -
Oh, why bother? It's obvious he knows this isn't fly fishing, isn't it? Or fishing of any kind? You don't fish with lute bodies. The "Flying" part of his jape refers to the experimental craft he took out of Rich Labs. It has a dual exhaust pipe, and no apparent engine. Is it venting the products of combustion? Radiation?
How does this anti-gravitation machine work? Obviously it hovers in place. Perhaps the radiation kills the fish, and the strange rods hook the corpses for easy retrieval.
Perhaps a day later the bird fell dead out of the sky. and the fishing man came down with terrible, unexplained blisters on his skin.