This could actually be really interesting.

Would there be regular animals alongside these critters, or would all the wildlife be 'monsters'? How are 'monsters' different fundamentally?

I do like the soul-binding thing. It has some very cool potential. It could also explain why, after you beat up a monster, it suddenly listens to you. The soul-binding process actually could have a taming effect. It could also provide a justification for why you faint if you lose... Though, I feel like 6 is a lot of customized mons to keep track of at once. Lots of record-keeping, status effects, and such, you know? Maybe commoners can have one mon, and 'trainers' can have, like, 2-4? Maybe there's a level-based limiter of some sort. I dunno. Also, I think there should be some non-soul storage method. Maybe after the training revolution, the humans figured out you can put monsters in gems of some sort?

As to the mentioned primitive monster-society, it could well be primitive for a lot of reasons. Maybe they're not inherently social, or their communities are isolated. A lack of communication and connection would slow technological progress. Maybe they are advanced, but it doesn't look like it from the outside. Maybe they just can't develop certain keystone technologies because they lack the materials in their native habitat. If you want to limit players to humans, maybe humans are unique in that they're the only creatures that can tame/train?