The Arcane Statistics page for each character has a line item labeled "Fishing Luck." What, if anything, affects this? Mine has always been at 0, and I've never seen any gear or potions which boost that stat.
Emberbright Cod is the only item I know of that has Fishing Luck anywhere.
The Emberbright Cod in theory, can be caught anywhere but have a high consistency of being caught in the Orden Mines. To be honest, a lot of people have the same question as you. I would help you if I could but there is no "clear" explanation of to why there is the luck section in your stats. In torchlight 1 there was the option to craft the Sushi Helmet, which granted some luck in fishing. One of the items used to craft it is Sushi Grade Fish Meat, which has been implemented in Torchlight 2 getting many people wondering. In the end the only item known is the Emberbright Cod, which is extremely rare and requires many many hours of fishing.
There are only a very small number of items in Torchlight 2 that claim to improve Fishing Luck and they are very rare. The problem is that there is no item in the game that really does so.
There are two reasons for this - first, in Torchlight 2 a bug in the affixes means that any items claiming to improve fishing luck do not do so - they improve magic finding luck instead. Second, there is actually no need for improving fishing luck in Torchlight 2 because none of the fish are classified as magic items, they are all classified the same thus there is no 'heirachy' of items meaning in turn improving your luck makes no difference in fish quality you find as they are all effectively just randomly rolled form the same table.
My Unearthed Arcana mod fixes this and adds the fish from TL1 meaning that not only does fishing luck work properly again, theres also a use for it.