Not only fighting games do this. It's a technique used in most games & genres. To name a few: Mario, Sonic, Zelda, Contra, R-type, Ghost-n-goblins, Out-run, Starcraft 2.
All those have several "tilesets" and "backgrounds".
The first reason, as PhilCK has mentioned, is that players get bored really when they get presented with the same art again and again.
The other reason is that art (even on non-interactive elements, such as backgrounds) adds another layer of meaning to the levels it is used in.
Take Super Mario's Ghost House, for example. The shadowy black tiles & closed windows in the background (and that music!) really do transmit something completely different from the ones in Yoshi's Island.