Different licensing deals work differently. It all depends on what's in the contract.
In the case of Donkey Kong Country, it was probably still developed by Nintendo, with Rare just being the developer. I'm guessing it's not too uncommon for publishers to get all the rights to the stuff that goes into a game, and probably especially common when the game is developed around one of the publisher's IPs.
Also, in the case of Donkey Kong Country, it makes sense for Rare to give up the rights to the other characters since they're all part of the Kong family. It'd be hard to use a character like Funky
Kong without Nintendo's permission.