"Any Jew will tell you that also?" Make that "any Jew who never learned the Jewish laws governing who is a Jew. Any Jew who believes in the Orthodox Jewish religion will tell you that Jew is a nationality. If Jewish referred only to religion and not to nationality, most Israelis -- including many of the founders of the state, and all of its prime ministers for the first 30 years of its existence -- would be non-Jews. According to Torah, if your mother was Jewish, you are a Jew, whether you believe in Judaism or not. (Conversion is a separate matter not relevant here, except that the same rule applies to children born to a woman after she duly converted to Judaism.)
You can find these laws in the 4th chapter of Tractate Kiddushin.