I don't think there is any evidence of fish, any fish, hide their food so they can eat later. We cannot generalize what we see in land animals to waterborne animals. Food in sea water will spoil very quickly. That is the reason fish, unlike land animal, never store their food. They cannot. We do see that damsels tend to their garden and chase other herbivores away.
Clown fish do bring food to feed their anemone, the larger pieces that they cannot eat, or when they are full. Larger clown fish, Clarkii and Maroon, bring smaller fish and feed the live fish to their anemones. We observe this in our aquarium. This behavior is advantageous to the clown and anemone pair, so they evolve and retain this behavior.
In aquarium anemones will do fine with out clown fish, if you provide the right environment for it, and clown fish will do just fine without anemone.