Racing Lagoon for PSX.
It's a hybrid RPG/racing game from Square, with really awesome music, about 50 or so cars that you can customize with thousands of combinations of parts, and a story that I wish I could understand more of since it's all in Japanese. Someone should do a translation patch for it, I think it would be interesting.
This is the main character, Sho Akasaki. You play as him, and you start with that car in the background, a Toyota AE86 Levin. In game all the cars are given different names to avoid manufacturer licensing, so the Levin is called 86-Lev (the Trueno which everyone knows from Initial D is called 86-Thunder)
The overworld map is like Final Fantasy VII with cars. You drive around and talk to people, and if you're caught in a passing racer's headlights...
You get to race them! The controls are very arcade like, you have to drift a lot with the handbrake and the only downside is that car collisions are pretty unrealistic, hitting a car from behind bounces you backwards a few pixels but opponents' cars are affected the same way, meaning that you can block them easily
Here's where you customize your car. After you win races, you can take one piece off an opponent's car (if you have enough RP or "Reward Points") and then put them on yours! Leveling up those 3 core components of your car gives them more spaces to put parts in.