Identity consists of multiple layers: nationality, ethnicity, class, religion, gender and various other factors. Concerning what makes someone a Jew, as far as orthodox Judaism is concerned religion represents an integral component of Jewishness, which is indeed why Messianic Jews are often denied permission to emigrate to Israel as they are no longer deemed Jewish. This is nonsense, of course, because one can be Jewish and an atheist. There are various other components which shape Jewish identity, for example, culture, history, language, and so on.
An important component of Jewish identity is, of course, ethnicity. So Jewishness is not just religious, it is also ethnic. Yet some (for example Iran's Ahmedinajad) seek to delegitimise the Jewish state by claiming its inhabitants are not Jewish at all, but rather European. But an interesting article in the Daily Telegraph details how new DNA technology in fact proves the Jewishness of Ashkenazi Jews, indeed even more so than some Sephardic lineages.
But what really interested me about the article was the reference to a fifth of all Spanish male lines having Sephardic origins. For those who don't know, in the late fifteenth century all Jews were exiled from Spain. Most moved to North Africa, but some remained, either converting to Christianity or else pretending to. The Spanish Inquisition was charged with rooting out false converts, which it did so efficiently and cruelly. As you know, I was brought up in Spain and I recall how in the early 1970s the local shopkeeper, hearing my father speak about the Jewish people, invited him to his home and showed him a secret room filled with all manner of Judaica. In fact, this Spanish family were Jewish, managing to keep their identity secret for five hundred years, which is quite incredible.