There are a couple of games where you can kind of 'play cars.' One is Simpsons Road Rage, where there is a mode that lets you explore the whole city of Springfield freely without having a time limit to pick up and drop off passengers. This is neat because there are a lot of funny gags you wouldn't notice from a regular playthrough. Like the movie theater is showing titles like, "A Popsicle Now."
Another one is a PS2 game called Road Trip. It's actually a Choro Q game in Japan and lets you travel around a world freely between races. It really feels like you're on a road trip!