I have to imagine the changes have to have been somewhat more driven by developers coming up with fun gameplay ideas while messing with new hardware. Super Mario 64 DS for example includes a collection of minigames with varying gameplay mechanics that explore different ways to use the dual screens along with the pack-in metroid demo, the Wii U had Nintendo Land, and the 3ds faceraiders, Mii Plaza and some AR demos, which all seem like polished proof of concepts for how to take advantage of the new hardware.
And fostering a culture at the upper echelons of management that enables, encourages, and embraces that mentality - that's the magic. Decades of fun goofy shit, and it hasn't yet been eaten by the business turds.
Nintendo also has a culture of sharing institutional knowledge regarding game design. Shigeru Miyamoto is not very involved in the day to day of Mario and Zelda development anymore. Other designers and producers who have worked with and learned from him have taken the reins of those franchises and taken them to new heights.
Contrast that with, say, Sega, who never found their footing with the Sonic franchise ever since the late 90s. Sonic games were developed by whatever developers happened to be available -- sometimes American or mixed Japanese-American teams; and the design ideas from the first few games that made them so great were lost once new hardware generations came along that couldn't use the original engine code. Sonic went a whole console generation (fifth generation, Saturn era) without a new full game in the main series, and the games featuring him after that have been a real mixed bag. (No, Adventure 2 is a mess. The controls are some of the most slippery imprecise shit in the 6th generation.)
