This is a bit of a technicality as it’s a game where player’s race through a side-scrolling platformer at the same time, but don’t face each other directly.
Track and Field games came to mind to me too, but I decided not to include them because 1) there are too many and b) they tend to be boring and c) they tend to be just one or two events in a collection of other events, and I was trying to focus on games built entirely around racing/running
Not sure what it’s called in every region but Konami did some terrible Boot Camp game where you play as a military recruit running obstacle courses. And if you were good enough at the running events, you got to die at war.