I'll be honest, I'd be hard pressed to name many animals strongly associated with specific countries. Off the top of my head:

USA - Bald eagle/American bison
Mexico - Golden eagle
Canada - Caribou
Peru - Alpaca
Bolivia - Guinea pig
Russia - Brown bear
India - Indian elephant/Bengal tiger
China - Giant panda
Australia - a zillion species only native to Australia
Japan - tanuki (raccoon dog)/those macaques that hang out in hot springs

...yup, that's all I've got.