As performed on the David Letterman show: https://www.youtube.com/watch?v=CLQ7xYnyIBQ
You can see a few shots of the Cuica -- the drum(!) that makes the high-pitched squeaky monkey noises.
Thank you! I collect variants of the Wii Shop song on various platforms, its one of my favorite songs from gaming and certainly my favorite dedicated-to-a-single-ui arrangements of music in software.
The Wii interface was a high watermark of style. Nintendo was clearly riding high aping aspects of Apple style in the 2000s but adding all the audio personality made it their own. Others were also invested in making something unique for the TV as an interface paradigm (Xbox 360 Blades, PS2/3 "orchestra" themes). Most of that has died away for practicality (and maybe accessibility) sake now, to flat boxes with minimal animation and no sound.
I looked it up, and all the Wii system audio was done by the same composer that did Pikmin, as well as dozens of other core (but secondary) Nintendo IP over the decades.
Nintendo appears to make the musical elements of its work a priority and it's good to see it escaping the game console into live performance -- following in the footsteps of both show tunes and movie music..
And for fans of video game music: that song inspired this little bit from the 8-bit big band: https://www.youtube.com/watch?v=v6hJvSkVLXs