Why do they say that with every toy I give them?

I think that's the message it gives when their happiness is already at 10/10. You can check on the pets page to see, but if their happiness is already full then you won't be able to play with any toys with them.

Also if your pet has a minion equipped or is checked into the resort or inn, their happiness should stay full and you won't be able to play with them.

Ahh ok, thanks.

Well, I've been on for 9 years and only just found out why. I thought they were just being fussy.

There are no fussy pets in Subeta. If they're at all hungry they eat what you give them if they're at all bored they play with what you give them (and destroy it) and they don't demand you tell them stories in different languages like Ytivan or Omeni or anything, all books are good.