I was in Germany in the mid-70s and again from 87-91. Unless the Status of Forces Agreement has changed significantly since 1991, American forces can own weapons without regard to German laws. They can transport them to/from Germany when they move.
Personnel living in barracks must store them in the unit arms room but those living off-post can keep them in their living quarters.
I don’t think this policy is well-known by the German public and I often wondered what kind of stink that would raise.
It changed significantly around 2000. I left in 2001 and was one of the last people in my unit allowed to keep my weapons at home because I was leaving right about the same time the ban on GIs keeping privately owned weapons kicked in.