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 first time in a decade that I wasn't gunning for the end game, pillaging the beta test to determine the fastest way to level and get to the "good stuff," and tweaking my add-ons to skip as much content as I could to get there. I even read a quest or two, though I admit to using the option (still available, even in vanilla) to turn off the line-by-line scrolling of quest text.
Only really to expand on Gráinne's comment as regards the 'collecting' and 'roster development' sections - which are, as said 'distinct' although they are intrinsically linked - I would like to point you in the direction of the in-game Achievements > Battle Pets > Collect tab. There are a series of achievements called Raiding With Leashes, pretty much all of which can be completed via the auction house - if you have the gold/inclination - or by completing old raids.
The original World of Warcraft was released in 2004, and World of Warcraft Classic aims to recreate the state of the MMORPG before its first expansion, The Burning Crusade, was rolled out in January 2007. Previously, players looking for a “vanilla” version of the game were only able to experience it on custom servers such as Nostalrius, which Blizzard shut down in 2016 to protect its intellectual property.