Is Cohen Jewish

Answer 1

The name "Cohen" (and variants) comes from a Hebrew word meaning "priest". By Jewish tradition, "Jewishness" is passed through the mother, not the father, so a Jewish man named Cohen who married a non-Jewish woman would have non-Jewish kids with the last name "Cohen". This isn't terribly common, as Jewish men have historically tended to marry predominantly Jewish women, but it's certainly possible for someone to have the last name Cohen and neither self-identify as "Jewish" nor be regarded by Jews (especially Orthodox Jews) as Jewish.

It's also certainly possible for someone to be ethnically Jewish, but not adhere to Judaism as a religion.

Answer 2

The name Cohen is Jewish yes. It means a member of the priestly class.