Continuity mistake: When Sean says he doesn't see J.D. as a threat his right hand is gripping the handle of his coffee cup. But when he takes a drink he's gripping the whole cup. Not the handle. (00:19:40)
Continuity mistake: When J.D. marks on the wall, Its not the same mark when the Janitor says "You drew on the wall" the second time. The first mark has a curve near the top, The second one bends in the middle. (00:01:10)
Answer: No, it was a fake, they used a stamp.
It is real.