I think he does put on a persona, but I think it is a persona based on his real personality. He uses his show to mockingly unleash his own inner narcissist that he knows has no real place in his life except to make people laugh, but it's there.
Sarah's link to the interview with Colbert seems to indicate him being a P. He says himself that he enjoys the interviews because he gets to improvise. I doubt he's an I simply because he looks so energized when he's on his show. He obviously loves the attention.