MP-38/MP-40 - 32 round stick (same magazine as a STEN, BTW)
Yes, they are compatible magazines.
In 1940 the British copied the German Bergmann MP28 as the Lanchester for the Royal Navy and used a fifty round version of the Bergmann box magazine, which was designed by Hugo Schmeisser.
When the Germans developed the MP38 and MP40 they also kept the Bergman magazine as compatible with Bergmanns in police issue.
The designers of the Sten adopted a 32 round version of the 50 round Lanchester box magazine.
During WWII there were 20 round, 30 round, 32 round, and 50 round box magazines in circulation and compatible between the Bergman MP28, Lanchester, M38 and MP 40, and Sten guns used my Germany and Britain.