It's simple. The mushroom world (let's call it Earth-M) is parallel to Earth. Now, when Mario and Luigi went from the real earth (Earth-R) to Earth-M, they knew they were in another universe. A fake universe as compared to their real Earth-R. Get it?
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Weekly Frostbite Quote #6:
(Family is at dinner watching TV)
Dan: That guy kind of looks like Bill Gates.
Mike: His name is Richard, so that would be Bill.
Dan: No--
Melissa: No, that’s Robert!
Dan: It’s William, you idiot!
Melissa: Oh.
Dan: Gotta write that down. (exits to family room and computer)