Paper Mario is an independent character from Mario and interacts with him, Luigi, and various other characters from both the real world and the Paper world within the book during the events of the game. Paper Mario himself is depicted as Mario in the Paper Mario series. Paper Mario is flatter than Mario, and has his own world, where everything (except the characters) is made of paper and cardboard.
In the games, Paper Mario can’t talk, Paper Luigi can, however.
In Super Paper Mario, when he was knocked out, he was that way for a guessed 2 months. Tippi came and rescued him and seemed to be in Mario’s vision, since she said she was “looking everywhere for” him, an enemy.