I think the games already established Mario and Luigi were born in the Mushroom World.
The "real" world, seen in Mario is Missing, Time Machine, GB Golf/Tennis, etc. can be accessed as well somehow, likely the portal used in GB Mario Golf.
(And I liked the Mario cartoons, contrived as they often were.)
The Brooklyn sewer warp story was what I was raised on, I admit, and I didn't warm up to the idea of them always having been there for awhile. But Partners in Time made their Mushroom Kingdom backstory a bit cooler.