This trophy can be missed on a normal playthrough, if you're just here for the story.
This trophy would actually be acquired if you would go for the all Social Link trophy, as it is tied to the Hermit S.Link:
Hermit (Rank 10) requires you to fish up the Sea Guardian for an old man in the Samegawa Riverbed. He wants to see one before his time is up, but there are more things to do to set this up.
1) First, you'll actually need to unlock fishing. This is a minor side-quest not actually covered by the Quest system. If you talk to the old man at the Riverbed, he'll say he could help you out in fishing if he had a hook. This hook will be found elsewhere in Inaba, but you will need a dungeon gem and a specific bug. A dungeon gem can be obtained from fighting in the dungeons, and the bug can be obtained from one of the kids at the Shrine. The boy off near the exit will ask for a drink/item that you will need to return with to get a bug. You need a Tatsuhime Ladybug, as no other will do. Return to the Shopping District at night. The item shop turns into a bar at night and, should you Talk to the owner, she'll go through her "son" Akihiko being hungry. Satisfy Akihiko by giving the owner the bug to obtain the hook, return this back to the old man, and you will be rewarded with the fishing pole.
2) Fishing by itself can be a bit challenging. Luckily enough, there are a few fishing books that you can buy from the bookstore to help you out. This will increase the number of chances that you can fish, as well as aid you in your strength when reeling.
3) (After 6/4) The boy at the Shrine will tell you he lost his bug-catching net. You will need to help him find this if you want to fish repeatedly. At night, talk to the woman in white and she should give you the net to give back to him. He'll let you keep it and tell you that you can catch bugs at a nearby tree. This will allow you to start going after:
Two more things:
4) To catch the Sea Guardian, you will need to unlock Shichiro Beach and have the sea-fishing rod. Shichiro Beach is unlocked through driving your scooter further away from Inaba. As for the sea-fishing rod, you must first go to the beach and try to fish there. After that, return to the old man at the Samegawa Riverbed and ask about deep-sea fishing. He'll tell you that he'll give you one if you can prove your worth by showing him a Guardian (more specifically, it'll be the Guardian you can get from the river). The conditions for this fish are thus: rainy (to increase the water level) and using an Inaba Jewel Beetle. You'll know if you are baiting one if there are several ripples and a large splash animation surrounding the bobber. Remember this animation, as it will be the key for the Sea Guardian as well. Once you successfully catch one, show it to the old man and get the deep-sea fishing rod.
5) Fishing at Shichiro Beach does not require a weather effect to catch the Sea Guardian. It does require a Genji Beetle to hook, though. You have a rare chance of acquiring both baits for these Guardians rarely from the bug-catching event (and the chance is greater if you have both bug books read and hit a perfect swing). Despite it being the bait required, you will not see any mention of it when using the bait. Trust me when I say that it is the required bait, regardless of it saying lots of Meguro Tuna.
You can fish up the Sea Guardian prematurely and hold on to it when you get around to that Rank quest for Hermit. After that, you could use any extras to feed the cat (as there are two quests, I think, that require such) for extra points (rather than feeding them small fish like goldfish or trout).