If this is not a joke-
Wndow putty used to have asbestos in it sometimes. Way before '93. If there was a caulk that had stose in it, there is no way it would last until now. and if it lasted that long, I would put it in my own shower. if it were in caulk it would be non friable and bound mainly in the matrix. releasing fibers along the fracture line.
Even if the caulk for some odd hypothetical reason happened to have asbestos in it one piece in your nose would not be much of an exposure. especially if you don't smoke. I would also look here