I would assume it wouldn't have anything to do with party or stable slot. Rather than what you are thinking of- it being along the lines of "equip x equip set to the slot y pet" if i'm interpreting right?-, a case of being in the equipment inventory and having presets that go "if a, b, c, and d equips are in inventory, equip it to the currently selected pet".

Like, at the moment you have the selected pet at the bottom of the inventory screen, but that instead of the tedious "click the relic, click the pet's relic slot, click the armor, click the pet's armor slot... etc" you have an extra section with equip boxes and a "save setup" button+a "load x preset" button+dropdown?
or something

or maybe just a "move the equips that are on this pet to [insert other pet here]"