The quality of captured wild pets is random. As of patch 5.1, upon entering a battle the Pet Battle UI shows the quality of a wild pet (idicated by a color-coded border around its icon). If you're farming a specific quality, if you forfeit the match your team will incur a small damage penalty (roughly 10%). There is no accurate way to determine a pet's quality prior to the start of a battle.
Different qualities aren't limited only to wild pets—many rare World drops and special non-wild companions have had their stats boosted. For example, the world-drop dragon whelps (Azure Whelpling, Crimson Whelpling, Dark Whelpling, and Emerald Whelpling) are of [rare] quality. Check your Pet Journal to see the color-coded rarity of your non-wild pets.
It was the experience of the game that brought pleasure and joy, and that's what a lot of us want again. Not the numbers. When I play Classic I want to go through the deadmines with people I like and overcome a challenge together and see myself become stronger as a result. I don't want to play Classic so I can say "booyah! That number is exactly the same as it was 13 years ago!"