I'm one of those people who fails to see the connection between PSO and Monster Hunter. Might as well throw in Dark Souls if you're gonna get that broad.
As much as I love PSO... I still hold the belief that it's HARDLY the first hunting game. It's ties to Monster Hunter (which really is the original), are so small that the only similiarity they hold is that they have a hub town and real time combat. Even then the feel of combat from PSO and MH is drastically different.
I mean, it's pretty obvious to me that the framework set up in PSO was modified and expanded to become Monster Hunter. Just like Monster Hunter inspired Soul Sacrifice and Ragnarok Odyssey and Toukiden. But in any case:
The current scope of Hunting Action game being used for this thread is any action games or action RPG's with a mission based structure, party based co-op limited in scope, and character customization of some sort.