...I'm going with the theory that Buddy lost his tags and had a new one made up on the Greek coin while he was stationed in Greece.
In 1915-16, that coin would have been 30 years old and would have still been widely circulated (take a look through your change and see how many coins you have from the 1970's)
I am also assuming that if the make-shift dog tag returned home, so did Private J. Rowley. Possibly he was a British or Commonwealth citizen that immigrated to Canada after the war?
There are some great searchable databases available on line at the British War Office and the Gov't of Canada listing Honour Rolls from the Great War, Medal Recipients, etc, but I have not yet found a J. Rowley with Regiment (Service) Number 16097...
In Canada blocks of these numbers were given out to recruiters, so if it is a Canadian number, you can tell who was standing in line in front and behind on the day the fellow signed up. I think it is interesting that the numbers 16096 and 16098 appear in the Canadian database, but there is no record for 16097.
I have also just read that Regiment Numbers were sometimes changed, and that it was common enough for a fellow to have more than one in the course of his service.