So... What IS Sonic's Schoolhouse?

We've all heard of the iconic meme game "Sonic's Schoolhouse". But... what exactly IS this game?

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TIMESTAMPS:
0:00 Backstory
1:32 Gameplay loop
3:45 Multiplayer

MUSIC:
Sonic Mega Collection Intro Theme [https://youtu.be/8PWBzQH5iF8]
Options Screen - Sonic R [https://youtu.be/fJXNx-EZGro]
SONIC's Schoolhouse - OST #09 - REPORT [https://youtu.be/QETrsUS6pKs]
Shop - Mega Man 11/RockMan 11 [https://youtu.be/2Rp2sp_BIwA]
Mickey's Speedway USA OST: San Francisco [https://youtu.be/SDTbHSrUzf8]
SONIC's Schoolhouse - OST #05 - A.MID / C.MID [https://youtu.be/8Rw54cRBcq0]

Footage from NintendoComplete unless stated otherwise