I play against my son on Super Smash Bros Wii U. How can I maintain a running score for each of us?
It doesn't seem possible to associate a controller/character with a particular nintendo account as you can with Mario Kart.
Try the Names feature, creating a name for each of you and always fighting with those names selected. There should be a section in the Records that keeps track of stats for each name you have.
It should save some stats for each character, although I don't know about scores from each game. Like @Toomai said, I think the Names feature works, although I cant be 100% sure.
Also, its not possible to associate 1 character with a Nintendo account, but I'm pretty sure you can sync either the game or the system to one, which SHOULD give you scores.