I don't know whether this is true of console versions as well, but on desktops/laptops you can have as many quest fish in your inventory as you like.
However, as long as there's one or more of the quest fish o' the day in your inventory, you won't be able to catch another. Once you're done fishing, load up and carry them wherever you like — for instance, to the Angler's home where you've strategically placed a bunch of Ice Chests (could be any storage item but these seem most fitting).