Attn all: I am typing this from the Monroeville PA yearly Militaria Antique Xtravaganza aka MAX. I am here because I have a similar watch. I have met bobskis2 and we have compared watches & notes from our fathers -- both former CIC in post-war Germany -- cross-referenced to the internet archives, and to 2 Skorzeny memoirs, and other bios of Skorzeny.
We now know that bobskis2 has the STOPwatch, similar but more sophisticated than a wristwatch. And yes he does have the one and only STOPwatch that Skorzeny was given. Still missing, perhaps forever, is the 18k gold WRISTwatch and the ruby-encrusted pocketwatch given only to Skorzeny. We know now that bobskis2's father has the real stopwatch which as is stated above, came from his father who arrested Skorzeny when bobskis2's father served in Counter Intelligence Corps
In addition gold STOPwatches were given to the officers in the Gran S'Asso rescue of Sept. 12, 1943. They were engraved on the back identical and were also Chronographe Suisse. I have one of these. There are only 6 given out, from our research. It has a leather wristband and is too small to have been worn by the 6'5" or so Skorzeny. Photos in the autobiography My Commando Operations show that the officers were small, as pilots and tank commanders had to be. One picture shows all 6 officers together and another photo shows Skorzeny dwarfing a few other men on the mountaintop in a posed photo before removing Mussolini to safe haven.
Two watches have surfaced after 66 years. Whether anymore will ever appear is certainly impossible to guess, but seems incredibly slim.
If you would like to be in touch on these watches with either of us, send us private messages. There is much documentation and letters of provenance available in addition to research in archives and books to corroborate all this.