The history of the Mas 45 is a bit more complex.
The Mauser factory at Oberndorff was captured by French Forces on 20 April, 1945. The French Army had that factory resume production of K98 rifles and P38 pistols in August 45. (you will find numerous threads on the weapons produced during that period with and without French markings on this and other forums). They then ordered .22 trainers also, and Mauser engineers produced the Mod.45, which closely resembles the KKW, but with a rear peep sight as per French specs (the German models had a tangential rear sight). About 8000 of them were produced in Oberndorff, until the Soviets asked the factory to be closed in August 1946 , because the manufacture of weapons was banned in Germany. The French Army then blew up the plant, but had its equipment transferred to France.
The production of the .22 trainer was resumed in France, first in Mulhouse, then in Saint Etienne. The first hundreds were assembled from German made parts and have combined Mauser/Mas markings, but all the others were from French made parts, with only Mas markings. The two models are not identical: the Mauser is blued vs. the Mas parkerised, the Mauser barrel is slightly longer than the Mas one, and it is grooved to the right vs the French one to the left. A great number of them were manufactured in Saint Etienne (hence the name MAS for Manufacture d'Armes de Saint-Etienne) as trainers for the armed forces and even for the civilian market, well into the nineteen sixties. Anyway it is an excellent rifle, of excellent quality, although a bit heavy because of the barrel.