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 It is out of date, but still functions without errors. So you just need to enable out of date add-ons in the AddOn List located on the Character Select Screen in order for it to show up in game.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.