Yeah. They did. Sonic Drift.
http://en.wikipedia.org/wiki/Sonic_Drift_2#Sonic_Drift_2
It's very much like Mario Kart. But...I dunno. It..didn't really catch on.
Sonic also had a fighting game which is quite lulzy.
http://en.wikipedia.org/wiki/Sonic_the_Fighters
Honestly, Sonic 2 and Sonic CD are like my definitive favorites. Those were the absolute best in my opinion. Sonic 3 was cool, as was Sonic & Knuckles. And then there's Sonic Adventure which I think is the best 3D adaption of the series, and then its just been a downward spiral from there. I seriously don't understand how they fucked up their formula and gameplay after Sonic Adventure. It was truly their best selling Dreamcast title and got Sonic back in the light. They need to stop adding so many shitty gimmicks and get back to basics like in Sonic Adventure. Sonic Adventure's truly the best due to its straightforward nature that focuses on the unique gameplay of each character rather than the stupid new additional gimmicks they add each game. The levels were designed rather well too.