They are not being identified by religion. They are being identified by tribal allegiance, which is more akin to ethnicity. For one thing, many people who identify as Jewish declare themselves to be atheist. How can that be if Jewishness is a religious identity? Just look at the Bolshevik movement, for example, which was largely dominated by Jewish atheists. Properly understood, Jewishness is neither a religious nor a racial identity (since conversion to Judaism, albeit difficult in modern times, is still permitted). It is a tribal identity, within which allegiance to the tribe is promoted as a cardinal 'virtue'.