Plot hole: Season 5, episode 20 (My Lunch). One of Dr. Cox' patients needs a heart valve transplanted. Heart valves don't have to be transplanted from another person. They can be replaced by a metal valve, or the surgeon can create a new valve from a different part of the patient's body. There was no reason to wait for a donor valve when they had other options available.
Plot hole: In the season 5 episode "My Fallen Idol", it is a key plot point that Turk has a hard time hugging guys. However in the Season 3 episode "My American Girl" Turk openly gives another surgeon a hug after J.D. makes fun of him. It is not possible for him to have changed because it shows that even before they became doctors, Turk would not hug J.D.
Answer: No, it was a fake, they used a stamp.
It is real.