Basically the boxes next to every filter have to be yellow thus making it an ALL set to be able to target someone else's Chocobo. I propose the following solution:
In the filter settings change Pets/Minions to Pets/Companions. Minions have a filter of their own so there is no need for them to share a filter with Summoned Pets. Honestly a hindrance there if someone decides to have a Minion out in a battle.