Is it possible that your father who came to us in 1906 and remained here until his death never became a citizen?

yes it is entirely possible, some like my great grand parents still had the old world mentatilty, the less that people knew about you, the better/safer you were. The requirements for citizenship have changed over the years, and quiet possibly they were relaxed enough back then, that most people didnt question citizenship..so it never became an issue