It isn't a paradox, it can be both a religion and a race. For those that consider themselves Jews the religious component is obviously more important. I am ethnically a Jew and yet I am a Christian (not religious, more a tradition and paper thing). I don't consider myself a Jew but some would. For racists that prosecuted Jews the ethnic component was far more important.

Judaism also has an ethnic component as those that are descendants of Abraham are considered Jews. And yet, those that adopted Judaism are too.