I think what surprised me most about that scene (as I'm sure was intended by the director and writer) was that the soldier was born in America and seemingly had no ties to Germany aside from his parents or other ancestors. He was probably an American citizen before going overseas to fight for Germany.
Was this a common occurrence for German-American families? Are there any accounts of this happening?
Yes. According to The German Order of Battle - Waffen SS and Other Units in World War II by George F. Nafziger there were 5 US Volksdeutsche in the Waffen SS. There were probably a few more outside of the Waffen SS but I think we can safely assume that their numbers were small.
Edit: Most of these were probably first generation immigrants or from first generation families returning to Germany for one reason or another.