Judaism is just a religion, by definition. What you seem to want to get at is Jewishness. Of course there is a big overlap between the ethnic component and the religious one, but the terms should not be confused. To answer the OP's question, talking about religious affiliation is more simple and politically correct. Religious affiliation is generally a cut and dry issue, but how someone ethnically identifies can be a gray area. In the US particularly, the major historical racial groups of whites, blacks and reds kind of tripolarized the view of people into broader categories, squeezing out finer nuances. All people from Europe were pretty much seen as whites, even the Irish, Italian and Jews (who also would have had Russian, Polish, etc affiliation).
While the totality of Jews has a pretty small ethnic standard deviation, it's not true that they are all one people. There are of course the Ethiopian Jews (Beta Israel) and other small communities around the world that have separate and often obscure origins. The Mizrahim (Middle East Jews) are also visibly different from Ashkenazim (Eastern European Jews). Keep in mind that the Jewish religion is not a proselytizing one (i.e. they don't go looking to boost their numbers through conversion) the way Christianity and Islam are. This is of course no accident, it is by design, since their religious texts promote the idea of being a chosen people to be served and exalted by non-Jews. Judaism stresses the concept of pedigree and lineage (from Abraham onwards... even within Jewry the Cohens are considered a priestly stratum). Some folks do manage to convert to Judaism but it is not easy to get accepted.
Were you taught that Hinduism is just a religion and not a people? We could similarly say that Hindus are pretty much all from India, Nepal and Bhutan. We should expect non-proselytizing religions to remain confined to a general area and racial group of origin, especially if we are talking about a region that was generally poor and did not go out exploring and settling other lands.