In 1787, Emperor Joseph II decreed that all Jewish citizens of the Hapsburg Empire were to assume hereditary surnames for administrative purposes. Though locally they might continue their custom of patronymics, i.e. x son of y, for official purposes, they and their children would have the same last name. Except among the urban middle class, these names were simply ignored except for official business.
Having to assign surnames to hundreds of thousands of citizens in a short time explains the “one from column A, one from column B” nature of many Jewish surnames. This also means that while these surnames may have been assigned by Austrian officials and have a German etymology, the people who bore them may have spoken any number of languages other than German.
The -stein to “-steen” pronunciation, however, seems to have been a strictly American affair, as a 1983 New York Times article explains:
A pattern is seen by John Algeo, professor of English at the University of Georgia. “The German names are usually pronounced with an eye sound. Most of the Jewish names have had the American influence of the ee sound, as in the words weird or receive, particularly that ei after the letter c.” Professor Algeo notes that, in Yiddish, a sound change occurred, with the ei pronounced as a long a, as in stain, but changed in American-influenced Yiddish to ee.”
“The ending of stein, pronounced steen,'' he concludes, ''reflects an American influence.”
Each family — or even each family member — may choose how to pronounce their last name. The composer/conductor Leonard Bernstein started out as a Bern-steen, but in adulthood changed it to Bernstein. So if California Sen. Dianne Feinstein wants to be a -stein and someone named Goldstein or Rosenstein wants to be a -steen, then that’s how their names should pronounced, because, of course, they’re American names, not German.