Hi owlsight, if you are in the sims 4 you can sell instead of harvesting if it gets to be too much in your inventory. The fridge unfortunately does not store produce or fish.
Fish can be stored in aquariums or fish bowls. Aquariums come in cats and dogs, and spa day and hold about 6 fish. Fish bowls come in base game and hold one. Whatever state of freshness a fish is in when it goes into a bowl or aquarium is the state it is in when it comes out. Please note that if you leave fish in a bowl or aquarium when you move houses the fish will disappear. The fish will go from excellent to fowl if left in the inventory and will only remain fresh if put in a bowl or aquarium. This will not mean you can't cook with the fish nor change the monetary value of the fish. It will simply reduce the boost to quality of the food that you get from using a fish as a fresh ingredient.
for general organization of the inventory I haven't discovered any better way other than using the Icya trunks from University. The trunks each have their own individual inventories so you can place root vegetables in one and fruit in another (as an example). You can even tag them with labels so that the title you give them comes up when your cursor is hovered over it. The trunks can then be placed in your inventory. A word of caution: when moving, adding or removing sims from your household you may loose access to the items in your icya trunk, its a bug.