Rare during the Nintendo 64 era was a second party developer (developed by an external party, published by an external party), not a third party developer (developed by an external party, published internally by the console manufacturer). Prior to making Donkey Kong Country, the majority of Rare's games on Nintendo were third party games. The Battletoads games were published by Tradewest for example, but they had games published by other publishers as well. I'm not exactly sure of the details how Rare got access to the Donkey Kong IP to make Donkey Kong Country, but I think it was Rare's excellent use of the pre-N64 release SGI dev hardware to model the characters that garnered them favor with Nintendo. Given the landmark success of the series, a strong partnership followed.

Nintendo has always been fond of favoring second party relationships and working tightly with them to create either new IPs themselves, or expand on their existing IPs. HAL Laboratories is famous for Kirby, Super Smash Bros, working closely with Creatures, Inc. and Game Freak on Pokémon titles, and the Mother / Earthbound series, and was so close that their darling programmer and later manager Satoru Iwata became the first president of Nintendo outside of the Yamauchi family. Other strong second party development studios that Nintendo has hitched their horse include but aren't limited to Intelligent Systems (Fire Emblem), Retro Studios (Metroid Prime, though now a first party studio), and Argonaut Software (Star Fox).

Rare was just trying to make games before their eventual partnership with Nintendo grew. I believe I read somewhere though that the relationship started to wane, so when Microsoft started wining and dining studios to build their own second party studio, when they came to Rare, the company saw the dollar signs and promises of more creative agency and decided to hitch to another cart. Since then, the company has released some successful titles since then, but arguably none are as notable as during the heyday of their partnership with Nintendo.