I don't quite understand the purpose of those, or how they work? Does it have something to do with doing battles while the minion is equipped to a pet or something?

Minion battling was a planned feature that was once being beta tested by gold account users back in 2016, I don't know if the feature's going to make a comeback. The minion levels in the zoo were a part of that.

Yeah, it's not an implemented feature at the moment, gotta keep fingers crossed for its return in the future

Yeah, right now it doesn't really do much except you can level up the stats by battling but you don't earn anything else from it. It's an unfinished feature that just kind of exists in a state of limbo at the moment.

For sure! I would definitely like to see that happen, it sounds cool.