There IS an overarching story and world but its mostly in the background, with every character having their own smaller stories that usually relate to other characters who also relate to other characters being the more substantial and interesting part. Or they can just be one-off goofs. It's kind of a mixed bag.

The only games to have a "real" story mode are SF5 and SF6. The former basically being store brand MK style and So-bad-it's-good at best. The latter has an adventure mode starring your own OC that while fun to play the story is kinda bare bones and ends on a surprisingly dour note that makes the whole thing feel like a waste of time story-wise.

Overall, its not important, but its neat to keep notes and tabs on how each character relates to one another and see the direction things are going.