Mode 7 is a special hardware only mode for the SNES. Technically, it cannot be done without an SNES.
However, mathematically, the "look" can be replicated and Allegro 5 or any other library works the same way. It's just a coordinate transformation. The question then is, what kind of transformation are you looking for?
If it's simply Mario Kart, you're just using a single texture at an angle and using bill board sprites for the cars will probably work. Billboard sprites are just polygons that always point toward the screen. Older games (and many modern) used them for trees to speed things up.
So you've got the "track" which is a single textured polygon (technically two triangles forming a square) that's rotated and a polygon for each car. And depending on the angle between the car and the camera, you "turn" the car by selecting a different angle picture. Much like Wolfenstein 3-D or Doom.