As far as I can know the fish that direcly affects your character end up in the items tab while the items in the fish tab only affect your pet. I did a bit of testing today on the prismatic morey, frenzy fish and trufflesnout fish. Each fish affects separate abilities so they should stack up nicely unless they overwrite each other. The pet tab only displays basic information about the pet which makes testing the troufflesnout difficult. I did however test feeding different shape-changing fish to the pet and each one overwrote the previous one. The numerical statistics tied to that shape also switched to that of the new shape. I confirmed this by having two types of shape-changing fish in stacks and alternating between them to observe the change in statistics.
Then I assumed that if the buffing fish effects would overwrite each other considering that they affect different stats, troufflesnout and morey should remove the speed increase from the frenzy fish. So I fed my pet the frenzy fish which caused a noticeable increase in speed and then I fed it the trufflesnout and prismatic morey to it to see if the speed boost would be lost. As far as I could see the pet retained the speed boost and attack speed for the full duration during this experiment and even when I fed it a shape-changing fish. So I conclude that yes, the effects do stack with each other.