answer: Religion. It is NOT a race nor ethnicity because there are Jews of every ethnic group.
You can change your religion and no longer be considered Jewish.
You cannot change your ethnic background.
Here's an experiment: look up photos of Yaphet Kotto, Alyson Hannigan, Adam Sandler, Natalie Portman, Lauren Bacall and Harrison Ford
Are Jews one ethnic group? Obviously not - all of them were born Jewish.
Being born Jewish refers to Jewish law – there are responsibilities and privileges in Judaism and laws were established to determine who is a Jew and who isn’t. That does NOT mean being Jewish is a race or ethnicity.
For example: one cannot partake of communion in a Catholic church unless you are Catholic and have recently gone to confession. A non-Jew cannot read from the Torah scrolls at synagogue.
If you look at the DNA and blood of a Jew it is consistent with someone from the middle east, even if they are European. It is a sub-race of Caucasian, a culture, and a religion.
The ancient Hebrews were a race, Judaism is a religion and a culture. But not a entirely a race. If someone leaves the Jewish religion, they are no longer Jewish.
Judaism is the exclusive religion of the Jewish tribe. Before, Jews apply only to the descendants of Juda, son of Jacob. Among the 12 sons of Jacob (Israel), Judah emerged to become the mightiest among the tribes and it became a separate kingdom hundreds of years after the reign of Solomon. Today, the term Jew applies to every Israelite or have at least Israeli blood.
It began as an ethnic group descended from Abraham & Sarah, Isaac and Jacob (who was renamed Israel by the angel of God he wrestled with). Today it's a religious faith, an ethically based way of life. A person by their law is considered to be Jewish (even if they don't practice the faith, as long as they have not converted to something else) if his/her mother is Jewish. A person can convert to Judaism & it is treated as a form of adoption into the group, possibly taking on the names of Abraham & Sarah and the male is circumsized if he hasn't had it done previously. Incidently, for many years, they did not encourage conversions since their way of life they consider difficult - and during times of anti-Semitism, it was actually illegal to perform conversions.
More of a culture. I know jews who consider themselves nonreligious, but they follow the traditions of Jews (the food, the holidays, etc.) because their families have done it for thousands of years.
Judaism is the "religion, philosophy, and way of life" of the Jewish people. Originating in the Hebrew Bible (also known as the Tanakh) and explored in later texts such as the Talmud, it is considered by religious Jews to be the expression of the covenantal relationship God developed with the Children of Israel.