Lots of WWI 32 autos with regimental markings. My understanding is that German officers were NOT issued unit pistols. They were expected to buy their own and their own uniform.
Both, actually. The WW I situation is muddled; there are actually also a lot of Lugers with individual officer’s name engraved.
But by the time of the Wehrmacht, there was a clear separation between the officer’s personal pistol, to be privately purchased as part of his kit, and unit guns that belonged to the army. P08 Lugers and later P38s were not available for purchase.
In contrast to the mostly 7.65mm personal officers pistols, the Wehrmacht carried battlefield-sized 9mm Para pistols as part of unit inventory and generally issued them to a function, like Kompaniechef or Zugführer (Company commander/platoon leader). Magazines were handed out separately as needed. These pistols changed hands as functions changed and men were promoted or transferred. Depending on the branch, besides the pistol a submachine gun might also be assigned to the position.
Away from the front, and while off duty but in uniform, the officer carried his personal pistol.
Obviously, this system came apart as the Wehrmacht did, although looking at surviving paperwork it is amazing how the military bureaucracy continued to function late into the war as armies in the field collapsed. Even for the last desperate Volkssturm units there exist detailed records of guns and serials issued and such.