The polar bear mob still stands as a slightly worthless mob; all it really has is ambience for the snowy tundra. Reviewing this and other mobs in such a position, ambient mobs (along with mobs that do have dynamic functions and interactions) may be the game's future. However, I think there's still some room for the polar bear to be given more interesting ambience and even some if not many functions or interactions.
• Attacking Villagers: in the upcoming 1.14 update, villagers will have a village variant in the tundra, the same territory as the Polar Bear's. As polar bears in real life are highly carnivorous, aggressive animals, what if they attack a villager that gets too close? They wouldn't seek out villagers or villages like zombies and illagers, they would wander as they currently do, and only attack if a villager is maybe 2 blocks or so within its proximity. It would also attack if a villager gets too close to a cub, as when a player gets too close to it.
• Luring with Salmon: before you go killing a villager-attacking polar bear (something people are sensitive about), what if you could lure them with salmon (no other fish) away from the villagers? But, here's the catch, if the player doesn't feed the polar bear after 10-15 seconds, or if they scroll their hotbar to another item in hand and put away the fish, the bear will attack. This would add more life to the polar bear as well as targeting uncareful villagers. This could have yet another function as follows:
• "Snowplowing": during attack mode (when hostile), the polar bear could clear away snow-cover underfoot as it speeds up, opposite to the Snow Golem's trail of snow that it leaves. This could make for a useful but risky way to collect snow or snowballs, or better to clear constant snowfall off of a house, lawn, or other build if desired.
• Fishing: with wolves and the upcoming foxes hunting other mobs, polar bears ought to have their own display of routinely fishing for salmon. They should even be able to swim fully underwater or beneath ice as in real life. Polar bears could have some way to detect water blocks with air above them in order to have a safe escape route through ice before losing air.