So what did militaries typically do in wartime, when a passenger car had a black chassis frame and different other color chassis components, with the body color different? Did they just spray the body color all over the vehicle (though protecting things like engine, tires, etc. when they needed to repaint the vehicle entirely?

I think it would depend on the circumstances. The motor pool may handle it a bit differently than other facets of the military. If they didn't have a lift, the soldiers would likely paint everything in one color. At least, that's how it was in the 60s and 70s.

During my mission in Bosnia in 1995 I met a Bosnian general, who used a Austrian origin Mercedes Benz. Though basically a shiny dark blue car, it was repainted with the help of spray can paint in different colors: brown and black spots where sprayed at random over the original color. Apparently without any plan: just where the 'painter' saw a free place a spot was sprayed.
Even the country sticker at the back, showing the Austrian 'A' was still visible.
I presume that would have been the case in many situations!
Maybe, you remember a famous picture of a German civil car, repainted/sprayed with OD with a hand painted white star on the doors.

That also depends on WHOSE vehicle it was. For instance: General Eisenhower's 1941 Packard was painted in OLIVE DRAB with WHITE Stars on the front doors, whereas General Patton's 1939 Cadillac Limo was left in it's original color of BLACK, without Stars. Joseff "Pips" Priller's BMW 328 was RED, one of Goering's several luxury-Benzes was LUFTWAFFE BLUE.

The General Officers of the HEER (German Army), had pretty much a choice of vehicles, of which many were BLACK, PANZERGRAU, or FELDGRAU. If they used STEYR , HORCH or MERCEDES 1500s, these were usually painted in the camo-scheme used in their theater of operations, or in the specific time-frame of the war; i.e, PANZERGRAU overall til early 1943, and subsequently, DUNKELGELB, with or without OLIVGRUEN and/or SCHOCOLADEN BRAUN camo... Pre-war German vehicles such as Mercedes G4s were "oficially" supposed to be painted LIGHT GRAY, with BLACK Fenders and Running Gear. Hope this helps...