Schtick is the closest I can come up with, although it doesn't directly imply any sort of proprietary nature, but rather business done by a performer as described above. You could talk about an actor's "trademark _______" I guess but it doesn't satisfy our quest for one word.

I had a Jewish friend who gave me a funny description of the meanings of schlemiel and schlamozzel:

"The schlemiel is the clutz who spills hot soup down the back of the neck of the schlamozzel." i.e. One is a bungling idiot and the other is perpetually a luckless victim of circumstance. Sorry to overstate the obvious...