Not so many people want to realize that the movie does not take place in modern times and the characters are not modern animal species. Here I'll try to guess real possible species of Brother Bear characters.

The animal that helped me the most in my reseach was visible in the movie and very clear to define American Mastodon that inhabited North and Central America from about 3,7 mya to 11,000 years ago.

Humans seen in the movie are wantering hunter tribes of Paleo-Indians aka. Paleoamaricans, known of hunting the mastodons.

A size comparion of American Mastodon and Paleoamerican compared to movie screenshot. Pretty accurate.

Rutt and Tuke are most likely a Stag-moose,a giant moose-like deer from North American megafauna.

Now about the bears. The most common bear species that inhabited North America were Short-faced Bears (3.0 Mya – 11,000 years ago). However of bears in the movie resemble more a modern Grizzly bear. At fist I was considering Etruscan bears(the ancestors of modern Gizzly bear, while Short-faced bears are ancestors of spectacled bears), but remains of this species are found only in Eurasia and North Africa. Modern brown bear(Ursus actor) evolved from Etruscan bears about 6 mya. They endered Alaska about 100,000 years ago and moved south 13,000 ya.

So I think that Kenai, Koda and other bears are "modern" Brown bear, however sice Brown bears and Short-faced bears existed at the same time, I would suggest that Igor is a Short-faced bear because of his different appearance and "foregin" language no one else understands.