Actually in most movies the watch has been issued by the government agency. If they are both spies, could make sense they both have the same watch. I don’t want to know more about the movie but saying this could be consistent with storyline.
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Dont encourage them. I can imagine the dialogue
"Q, I need a new watch, it needs to be very strong but also light and very accurate"
"youre in luck, Bond speaks very highly of this watch, unfortunately, only licence to kill agents are issued these on bracelet but I think youll find this Nato strap is both durable and comfortable, im particularly fond of the coffee coloured stripes. Speaking of coffee, I find this krups coffee machine to be compact, ergonomically designed and has variable strength setting for those long night missions"