These are simply random events. So, yes, their occurrences are randomized.
Feebas will randomly appear in your aquarium.
Just like Feebas, Manaphy will randomly [appear] in your aquarium.
According to Bulbapedia, both of these events can randomly occur during feeding, but this is the only source to say this. All other sources simply say it occurs at any time while at the pond.
According to PokemonGoHub both of these events can randomly occur at any time while viewing the Home screen:
Randomly occurs when viewing the home screen
This is also explained in this IGN article:
Can occur anytime at the pond
In order to trigger the bonuses these events provide, you must tap on the event Pokémon.