I can only speak from a UK Military perspective (British Army 1993 to 1995, Royal Air Force 1995 to 2005).
Casio G-shocks were never 'issue', but were widely available at NAAFI's and PX's. I believe the DW6600 briefly carried an NSN (NATO Stock Number) but this was for the convenience of the US Navy, who issued the DW6600 in small quantities to divers.
Most people in my unit kept a G-shock as a 'work-watch' (privately bought at a heavily discounted price).
It was common practice (UK) to etch the backplate with country, surname, initial and last 3 digits of their service number (see photo). I think US troops would often do this too.
This was not really to prevent theft, but in the event that you were either captured or killed in such a manner that your body was not recoverable, it was likely that items such as weapons, webbing and wristwatches would invariably be stripped from the body and stolen. By 'marking up' possessions, it might give investigators the tiniest clue to follow if later discovered 'post-conflict' (this being at odds with Special Forces, who would often 'sanitise' their possessions to remove identification).
Crude etchings like this is often an indicator that the watch is "ex-military", however apart from the US Navy, I don't believe they have ever been "NATO issue"....
