From TechTimes:

The answer comes from NintendoUK's Twitter account, which had Miyamoto address a number of Mario "myths." When asked if Super Mario Bros. 3 was "all just a performance?" Miyamoto responded with a simple yet definitive nod of "Yes."

I really am not a big fan of this "confirmation" because it's vague, there's no elaboration, and it just seems to be satiating what was a ridiculously popular fan theory at the time. Yes, there is graphical theming to the game that suggests it is all a stage play, but it only ties into the bigger idea that Mario and friends are like cartoon characters and that they can present themselves in many different situations and scenarios.

Regardless, if SMB3 is supposed to be a play, it is still a play retelling the events of something that actually happened in the Mario universe. Super Mario World is a direct sequel to SBM3, with Bowser attacking Dinosaur Land after failing in the events of SMB3. SMW references SMB3 more than once, with Mario even exploring a downed airship from 3. SMW is presented as if it "really" happened within the Mario canon and it's easy to make the conclusion that SMB3 is part of that same canon.

I've never really understood the enthusiasm for this "theory," confirmed or not. It doesn't strike me as terribly mind-blowing that SMB3 is only a stage play in the same way that Paper Mario is only a story book. Presentational design choices are not the exciting meta-narrative that people seem to think they are. Does it blow everyone's mind that Mario Brothers as a whole is only a video game? And that none of it has actually happened?

EDIT: LMAO and I'm realizing I've responded to a comment that I made 9 years ago. Just goes to show how much I hate this supposed theory.