I have always wondered whether or not the Paper Mario games were really set in a stage or not. I have come to the conclusion that they aren't. My proof: in Paper Mario: TTYD, the battle scenes are set on stages with an audience and dangerous props. HOWEVER, in the very beginning of the game, before the title screen, we find out that a storybook opens. Inside is the adventures of Paper Mario. It's like the player is reading a book on the past events of Mario. He is paper because it is being re-told in the book, and the pages are made of paper (obviously). Think Yoshi's Story.
My second reason: Even though there are deadly stage props and an audience in PM: TTYD during the battles, remember that the audience is also paper. And also, when Luigi is in the audience, he sometimes tosses items to Mario to use against enemies, which implies that Luigi is afraid that Mario may die, which further implies that this isn't a play, but is in fact an actual battle in Mario's history. Not to mention that the bulky bob-ombs in the audience can exlode and kill everyone in the audience. And Hooktail even eats half the audience during the boss fight, and gains HP from them.
So...my conclusion is: The Paper Mario games are set in the Mario universe. The stage is only used to make the battle system more unique.