Y'know, part of their police procedures is that you have to be wearing an identifiable police uniform if you're carrying a gun. Since Adachi is a so-called plainclothes detective, he wouldn't actually carry a gun on him. It would stay locked up at HQ and he would have to get permission to bring it with him on cases.
Like, remember how he didn't just whip out his service pistol during Rise's stalker / photographer incident? He wouldn't have one on him to begin with, and he wouldn't try to get permission if he thought he was just going to babysit Dojima's nephew. One would assume he has a gun in December because it's one of his model guns that he rigged to fire, or he just took it from the station without anyone knowing (Dojima had been hospitalized since November after all).
In December, he has a line inside of the TV where he says he became a cop so he could legally carry a gun. For a little context, you can't own a gun there as a civilian. We do know from P4U2 that Adachi does like guns and has various model guns that he does maintenance as a hobby on 'cause he can't take his service pistol home with him.
But, even knowing this, what he says in December feels a little nonsensical. Cause at some point in his career, he decided to become an investigative detective in a country where you don't carry around any sort of weapon as an investigative detective. Idk, I just can't take him seriously when he says that lol. He even says it in the TV world when he's in "B-movie bad guy mode" and trying to make himself out to be the Worst Person Ever.
In contrast, the uniformed Officer Kurosawa in Persona 3 would have a gun. Actually, I assume the black/gray object on his right hip - that you really have to squint at this screenshot to even notice - is his holster. He's stationed at a satellite police station in the mall and would be expected to respond to incidents that happen in the area, which may or may not involve using a baton or gun on a suspect.
Soejima's rough drafts reveal something about this topic and Atlus's process.
In Soejima's draft of Dojima, we see that Soejima did draw him with a gun holster on his left. But in Dojima's finalized full body character artwork, this magically vanished. Soejima - or whoever gave him specs for what Dojima's design needed - might have assumed that plainclothes detectives do carry guns but this later got corrected and we see this reflected in Dojima's finalized artwork. Adachi's own art is pretty useless for this cause he's always wearing a jacket, but I found this notable regardless cause Adachi and Dojima do share the same profession.
Edit: If you are wondering why Naoto has a gun, I have no idea lmao. That seems more about Naoto being a party member and needing a weapon for combat than it does any kind of basis in reality. Like at first we only see her with it inside of the TV, so it just seems like "a TV world thing". But I've been told that in Persona Q it's revealed to be an actual real working gun? Welp okay boss if you say so.