I know the cat/dog, moogles too.

Are there any other pets that would interact with each other?

I saw the mandagora queen tooltip and just got curious.
It sure sounds like something could happen.

"Though wed to the Tomato King in a move to strengthen the bond of Houses Tomato and Mandragora, the queen longs to steal away with her one true love─the Eggplant Knight. What will become of this forbidden romance!? Use item to acquire the Mandragora Queen minion and find out"