I agree that Jew is not a "race". Jewish is a religion, although there can be cultural features that develop out of religious practices that then form communities of people.
"Race" is a social construct used to organize people and resources. "Race" references phenotype (skin color, facial features, hair texture). People and resources are valued and organized according to notions of "race".
People of various "races" can be Jews. For example, I know "Black" Jews in Canada, and there have been media reports of "White" Jews in Israel discriminating against Ethiopian/"Black" Jews.