Not to be pedantic, but
2 big bears, 7 types of cervines, 2 goats, 2 big cats, An ox and a bison and 2 pigs.
2 big bears- grizzly is a big bear, and brown bear, but they are the same animal. Black bear are pretty small.
7 types of cervines- cervids, deer family, include moose (3 types), caribou (3 types), elk (3 types), deer (6 types I can think of off the top of my head)
2 goats- there are no goats (Capridae) native to the American continent, but there are Mountain goats (Antilocapra). They are a unique species, as are Pronghorn antelope.
2 big cats- Mountain lion is certainly a big cat. Jaguar is big but is protected. Ocelot is also protected. That leaves the small cats (lynx, bobcat, margay maybe) that can be hunted.
An ox- ?? drawing a blank on this one?? Cape buffalo are a true ox.
a bison- Wood bison and plains bison.
and
2 pigs- 1 pig and a peccary (if you are referring to javelina). All pigs are the same genus and species (Sus scrofa), whether you hunt Russian boar, ridgebacks, or feral pigs, they are all the same thing.
All in all, I want to hunt with Buzzcook! Looks like he is going to be having some fun!