I must have missed this game during the late 90s:

It is the intro of a game called Shutokou Battle Gaiden: Super Technic Challenge created by Megaquest in 1996. It must have picked up the drifting craze in Japan during the 90s and created this game for the Japanese PSX. Later on it had successors on other platforms as well.

Currently I'm playing old games at work on my old trusty PSX during the lunch breaks, so I definitely want this one! :)

Update: I did find Shutokou Battle for the Saturn and the Shutokou Battle for the Super Famicon (SNES). But still no PSX version yet...