I think he was asking about Jews. Not about Judaism. There seems to be some confusion due to the fact that there is more than one definition of "Jew".
Wrong. A Jew is the one whose mother is Jewish or who has made
true giyur (conversion). This is the
ONLY true definition.
Any person can convert to Judaism and become a "Jew", but one cannot become an ethnic Jew by converting. For that, he must be descended from Jews.
The one who converted to Judaism has become real Jew, without any quotation marks. On the contrary, the concept "ethnic Jew" is misleading.
In the old days, this distinction was taken very seriously - to the point where R. Yehudha haLlewi, in haKkuzari, tells the king of the Khazars that though he may convert, he will always remain a second class citizen among real Jews.
Rabbi Yehuda HaLevi knew that the king of the Khazars wanted to convert all his people to Judaism, which is forbidden, and wanted to prevent it. Mass conversions are not welcome in Judaism becuase there is no chance that all people in such group sincerely believe that Judaism is the truth. They either seek some benefit or they are forced to do so, such motives make the conversion phony and invalid. Khazars didn't truly convert to Judaism, they didn't become Jews.
Even Ribbi 'Aqiva, who was descended from converts, was not considered on the same level as true ethnic Jews.
Of course there is such concept as Ancestral Merit ("Zechut Avot" in Hebrew), and a convert doesn't have it. But it has nothing to do with the definition of Jew. Moreover, Rabbi Akiva was a great Talmudic sage whose Halachic rulings "true ethnic Jews" follow until this day!
No, ethnic Jews are not a race - but we are associated with the Middle East and we are, therefore, of Mediterranean type. DNA studies have (reportedly) shown a strong genetic link between ethnic Jews in most of the world.
Not DNA but unique Jewish soul defines Jew. And this soul is acquired only either by birth from Jewish mother or immediately after true conversion. And even the Ancestral Merit, which a convert doesn't posess, is purely spiritual concept. DNA is irrelevant.