Without using a Mod the only way to keep fish fresh in excellent quality is to immediately put them in a fishbowl or aquarium. If you're fishing in your own neighborhood, that's not a problem. What I'll do in that case is pause after my Sim catches a fish, click the 'center on lot' button and open the fish tank. Drag the newly caught fish from your Sim's inventory in there and then unpause so your Sim will begin fishing again. Actually, you really don't need to pause since this can be done while they continue to fish.
The biggest problem with this method though is the maximum fish allowed in a tank is I believe six and fishbowls only one. If you want to store more than six fish, you'd need more tanks and they're not exactly cheap. You could just keep six and just fish again to refill the aquarium as they're used to make meals though and I've actually done this myself. However, this won't work at all if you're fishing in an area outside of your neighborhood. Makes no sense to suffer through a loading screen every time your Sim catches a fish to put them in a tank. Especially if they're on vacation.
There's one other thing you could do to store freshly caught fish in neighborhoods outside their home area, but it's a bit ridiculous. I haven't tried it, but it might be possible. Put down a park or any kind of venue in a neighborhood where you want to fish if there isn't one already there and place some aquariums on the lot. Use this lot to load into the neighborhood, for instance perhaps use the museum in Willow Creek to go fishing in the public area behind it.
I think in this case you'd be able to put your fish in those tanks and then just retrieve them before ultimately going home after they are done fishing. I'm not 100% sure you'd be able to do this with aquariums located in venues though, unless your Sim happened to own the venue, such as a retail store, restaurant, or Vet clinic. Plus, you'd still have the problem of the fish going foul before using them unless you have an aquarium at home too.
I've been campaigning for some kind of fishing cooler, an update to the coolers we have from OR and I believe the Toddler pack has a cooler, or a home freezer to allow the storage of fish for a long time. Every time someone posts a thread about what would you like in the game, l try to remember to mention it. So far, no go, but I keep hoping! ;)
Happy Simming! <3