No not at all.
The reason you see stable masters is because they "stable" additional pets.
Some hunters have more than one pet, depending on what thy want to do, like UO in a lot of ways.
So for a fee, you can buy storage and save your pets. LImit of two slots beyond what is standing next to you I believe.
Otherwise when you logout, have your 'happy' pet next to you and all is well.
Or you can dismiss it and whistle for it when you need it.
Bo