Yes, a lot of Jews did change their names when possible, especially the convert Jews, though before recent times the procedure was usually pretty complicated, but if someone in interwar Germany did had German Ashkenazi name like Goldberg, which were specifically assigned to Jews in order to mark them as Jews, it’s sure that someone in straight paternal line was a Jew, though not necessary father.

But what i meant is it would be rather impossible to live back then, have Jewish name and not even be interested about his Jewish roots?