The Paper universe is gone
Mario & Luigi & Paper Mario confirms that there is some kind of multiverse stuff going on with Mario. The events in each universe are similar--Bowser kidnaps Peach again and again and Mario saves her, etc. In the Super universe, Super Mario was traveling through space collecting stars to save Peach from Bowser. At the same time in the Paper universe, Paper Mario was traveling through dimensions collecting hearts to save everything from melodrama. At the conclusion of Mario Galaxy, I guess the universe is destroyed and gives birth to a new one and Rosalina explains that it's not going to be exactly the same (apparently. Sure, I still don't see how this is supposed to be what happens, but it'll serve my purposes here). At the same time, the Chaos Heart is destroyed and the worlds the Void swallowed are pulled back out.
...But they're not exactly the same there, either. Paper Mario's home dimension was one that the Void sucked up, and the Star Spirits didn't have a more recent System Restore Point for the world than one before Mario beat the Shadow Queen. She became unstoppable and took over the world before Mario could collect enough Power Rush badges to prevent it. The world Paper Mario and his friends call home became a living hell.
Sticker Star is what happened when they escaped and sought refuge in the Super universe--the Super characters, not comprehending that these paper counterparts were more of thinking, living beings with souls than anything they'd ever met, put on a puppet show using Paper Mario and co.'s bodies. Paper Jam is the Paper cast's escape from the Super universe into the M&L Universe, where once again they will be used as props simply because of their strange physiology. It is possible they could escape to Woolly World and eke out a meager existence there next...
...but they'll never really be at home again.