You are correct in that those convicted of DV Assault, whether a felony or a misdemeanor, are prohibited from possessing a firearm. It is possible that he purchased the weapon before the conviction and never surrendered it upon conviction or perhaps he obtained it from the private party and no background check would be required.

I would presume that APD checked it, while in their possession, to ensure that it was not stolen. It would be highly unfortunate if APD returned it to him, after his DV conviction, without checking to ensure that he could legally receive it due to the DV conviction.