Nintendo did technically create the DK series, in that they made a series of games with "Donkey Kong" in the title, but none of them are really part of the modern DK series. Donkey Kong is a proto-platformer, really part of the Mario series, as a precursor to Mario Bros. (Donkey Kong and Donkey Kong Country have about as much in common as Yoshi's Island and Yoshi's Cookie); Donkey Kong '94 is also part of the Mario series, and, like the Mario vs. Donkey Kong subseries that it's retroactively part of, it's really a puzzle game. Donkey Kong Jr., Donkey Kong Jr. Math, and the two-player mode of the Game & Watch version of Donkey Kong 3 are the only games in the pre-Country, Nintendo-developed DK series where you can play as a Kong, and only DKJr. even slightly resembles what we would now call the DK series (I'm not even gonna bother with Donkey Kong Hockey and the like).
When Rare started the Country series, they created an entire new universe. And the basic elements of that universe, none of which existed before Rare came along -- jungle setting, collectible bananas, other members of the Kong family -- are firmly entrenched enough in the nature of who DK is that they show up in the only non-Rare games in the modern DK series since then: Jungle Beat, Konga, Bongo Blast, and the Clu Clu-alikes (not to mention more than a few allusions in the MvDK series and especially in recent Mario sports titles). For all intents and purposes, Rare created the DK series as we know it today.