I sort of disagree with iain, but this is no big deal, since where there are two Jews, there are three opinions.

I'd say that Jewishness is an ethnicity, and that Judaism is a religion. The two overlap, but are not identical. Members of Beta Israel identify themselves and are recognized as members of either or both categories.

As for the concept of race, I don't see it as relevant to the nature of Jewishness, since the only racial category that has any precise meaning is "the human race."