Is Stein a Jewish name?

No, It's a German name meaning stone. You can be Jewish with a German last name. Note that some people who move to a new country don't keep their old surnames and they pick knew ones or their named after their occupation. My mothers family on one side were trappers so they were named Fanger (meaning trapper or to catch). I also know of some Jewish family's with the last name Roth and my last name is Roth but my family is not Jewish Roth ( meaning red in German ). People were named Roth usually because of their red hair. So basically not everybody with Stein as a last name is Jewish.