Interesting post. Not sure if I would hunt with it, but interesting.
In the US, it would be up to the states to determine if you could use this Air Gun to hunt with, because it is definitely not a firearm or archery.
In the US, 18 U.S.C. § 921 defines the term “firearm” as: (A) any weapon (including a starter gun) which will or is designed to or may ready be converted to expel a projectile by the action of an explosive; (B) the frame or receiver of any such weapon; (C) any firearm muffler or firearm silencer; or (D) any destructive device. Such term does not include an antique firearm.
The key term being "expel a projectile by the action of an EXPLOSIVE".
It would also not be consider as "archery", as the common definition of archery is "the sport or skill of shooting with a bow and arrows, especially at a target". Again the key term being "...bow AND arrows..." And this is not a bow.
I think most states would consider it an air gun. The common definition is, " a gun that propels a projectile by compressed air" Which is what this does. Now the question become what states allow hunting large game with airguns. Either way, this is interesting.