From the Super Mario Wiki
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Paper Luigi is a paper incarnation of Luigi based on his appearance in the Paper Mario series that came out of a magical book in Mario & Luigi: Paper Jam. Paper Luigi is an independent character from Luigi and appears only after the player unlocks the Music Player after beating Shiny RoboBowser, though he is mentioned by Starlow at the beginning of the game when Paper Mario first meets Mario and Luigi, and confuses Luigi with Paper Luigi. He appears to sleep in Twinsy Tropics while Mario, Luigi, and Paper Mario are on their adventure.
When playing certain songs in the Music Player, Paper Luigi has different reactions: when one plays a song relating to Princess Peach, hearts begin to appear as he listens to the song. When a song regarding Bowser plays, he is shown to sweat in fear.
Paper Luigi was planned to be playable in Mario & Luigi: Paper Jam but was taken out because the team thought it would be too difficult to control four separate characters.