Most Japanese fans call him (or possibly her?) "Oo Namazu sama" (Master Big Catfish, 大ナマズ様), as s/he did it in the dialogue of Meiling's Scenario of Unthinkable Natural Law.
The Giant Catfish has been nicknamed "Primeus" by some fans. This is due to some similarities between the catfish and a MUGEN joke character named Primeus, who was a giant floating salmon who shot laser beams. Said MUGEN character has since been modified so he sometimes uses the Giant Catfish's sprites during his introduction to battle.
"Giant Catfish" has been a recurring joke or theme in the English Touhou community since Scarlet Weather Rhapsody when Reimu seemingly brought up the subject out of nowhere (no such associations between catfish and earthquakes are commonly known outside of Japan), and when Sakuya caught one for a banquet.
Fan often compare the Unnamed Giant Catfish with Whiscash, a Pokemon that looks somewhat similar. Sometimes the Unnamed Giant Catfish is also depicted as Magikarp's new evolution.