▶ Day 3: Create a color porn coloring
Its real-life cognate, the Metropolitan Police, was established by a parliamentary act in the 1820’s and there was a conscious effort to distance them from the military—this is why the constabulary wear ties and bowler hats instead of a more military-style uniform, so they look approachable!
I don’t think the City Watch is a subdivision of the military, but they’re obviously neither a private enterprise nor a local initiative. Early police Commissioners were military veterans, and Admiral Havelock mentions in-game that he once sailed with Geoff Curnow, implying that Curnow has a navy background. Despite the similarity in appearance to Royal guards and hired aristocrats’ guards, the City Watch is not, I don’t think, a subdivision of the Empire’s military, but their own force in the employ of the local government. However, martial law is basically if not actually in effect while we explore Dunwall, so the Watch is most likely being supplemented by branches of the military in order to control the plague.
Also, considering Curnow’s place in court, and his meeting with Campbell as a representative of the City Watch in settling a dispute, I think it’s safe to say that Captain Curnow is in charge of the whole City Watch force.