Doesn't that only confirm that it's a play in the games where the visuals explicitly show that it's a play? Beyond that Mario games don't seem to clearly be in any sort of fictional medium context at all, just because SMB3 was a play doesn't automatically mean Super Mario Sunshine is also a play or a movie, at that point that's just the characters' lives.
I actually find it funny trying to make any sense of Mario mythos at all. The characters are endearing placeholders for protagonists and enemies and create an easy design language to use for a game, but there's not really a consistent lore, the closest we've come to this whole sort of idea being legitimatized is how Nintendo has handled Zelda.
They could eventually try to create a Mario canon, but I think it's a bit too silly of a franchise to try to seriously do that with, the characters and world just bend to fit whatever works best on a per-game basis, for tone and mechanical reasons more than anything else, whether that's a play or ostensibly real characters in their regular lives.