The add-on is Pet Battle Quality Glow. It is out of date, but still functions without errors. So you just need to enable out of date add-ons in the AddOn List located on the Character Select Screen in order for it to show up in game.
As seen in the screenshot above, Pet Battle Quality Glow changes the icon color of the pet you are fighting, indicating its quality. Just like items, blue is for rare, green for uncommon, white for common, and grey for poor. The add-on also changes the color of the name.
This macro, from forum poster Simca, enters the quality of the pet in your chat box.
/run for i=1,C_PetBattles.GetNumPets(2) do SELECTED_CHAT_FRAME:AddMessage(C_PetBattles.GetName(2,i).." - ".._G["BATTLE_PET_BREED_QUALITY"..C_PetBattles.GetBreedQuality(2,i)]) end
Make sure to bind the macro to a hotkey that is not 1 through 6, as the pet battle UI uses those keys. Though you won't be able to see the macro during the battle, it will still activate when you press the key you bound it to.
A way to check pet quality before battles would be more valuable, but I don't see Blizzard providing that information as it would take the mystery out of wild pet hunting. At least this should save some time during the pet collecting grind.