This might be a Spoiler if you haven't seen the movie but The bear is actually one of a set of 4 princes; in a story that Elinor told Merida; there were 4 princes that ruled parts of the land; but one wanted power; so he went to the Witch and told her he wanted a spell to give him the strength of 10 men, so the spell turned him into the bear.

He is a prince but he's not Merida's love interest

not anymore XD

If you had watched the Legend of Mordu, it shows that he was one of the 4 brothers of a fallen kingdom. So yeah, technically he is a prince