The Star Spirits got the real Star Rod from the Fountain of Dreams and replaced it with a fake one thinking the Dream Landers were too stupid to notice and the placebo effect would do well enough
And they were right
By extension, Popstar is out there, in the same reality as Mario's
Really, Nintendo's major franchises do a good enough job of being able to conceivably co-exist... Metroid, Kirby, Star Fox and F-ZERO all take place on other planets and Olimar is from another one. StarTropics and EarthBound both take place mostly in surrealistic fictional locales on an Earth with real locations mentioned (one man in Fourside mentions Japan, Mike Jones is from Seattle)--Pokémon could take place on this same Earth, seeing as Lieutenant Surge is from America and some 'Dex entries talk about India and stuff like that. Time periods allow for the aforementioned space-themed games, Zelda, and I guess Fire Emblem. Mario is actually the toughest one to place... it'd be dumb to use the "it's not Earth" explanation, but it is wacky enough to not be, and consistent enough about how its unique species are all OVER the place that it's hard to see it as being just a part of the same world as anything else. It's nothing to lose sleep over but I do like the Brand Universe concept a lot
Toads have giant brains inside their mushroom caps
But size doesn't matter. And if it did their brains have fungus all over 'em
Toads don't have bones