Religion, not ethnicity. I'm Jewish, and the old saying "two rabbis, three opinions" applies here too:
What Christcorp said is kind of correct. Basically, a Jew is someone who was either born to a Jewish mother (or father, depending on movement), or converts to Judaism. It's kind of hard to convert to an ethnicity!
That said, the belief that someone can be a Christian or Muslim Jew is not universally accepted among Judaism. Typically, Reform and Conservative Jews believe that once you start practicing another religion, you are not considered a Jew.
To answer your question, nothing I filled out this year listed "Jewish" under ethnicity. It's basically universally accepted on applications and by the government that Judaism is a religion and not an ethnicity.