Some people forget (at least, I certainly do), that the characters in fighting games have full backstories behind them as well as an overall reason for being in these fights. It's not just "well, it seemed like a good idea at the time." If you're looking to explore more of that, you'll want to check out the Fight! RPG. The second edition of the game is up on Kickstarter now.
From the campaign:
Fight! 2nd edition is a revision and enhancement of the best tabletop emulation of the characters, combat, and stories found in traditional 2-D and 3-D fighting games. This genre shares some similarities with superhero RPGs, as the main characters are powerful, possess powers beyond the limit of human capacity, and face off against similar characters in stories where the fate of their neighborhood, city, world, or even the universe is at stake.
In fighting game stories, there is a focus on martial arts and fighting spirit as the greatest sources of power in the setting. Fighting game characters live in worlds in which martial artists are greater weapons than high-tech armies, fighting spirit is a revolutionary source of power, and major events in world history can be decided by the winner of a tournament.
The mechanics of Fight! prioritize elements of the genre to enable each character to really feel and play like a unique character in a fighting video game. Even with this focus, the genre itself allows for tremendous flexibility to tell stories involving street gangs in a bad part of town, celebrity fighters as rivals on the world stage, black ops espionage missions, fantasy epics, space opera, or even cosmic battles between gods!
The Kickstarter's closing in on their funding goal with still 15 days to go.