Judaism is both an ethnicity and a religion, or an "ethnic religion".
Ethnicity: Affiliation or identity within a group of people bound by common ancestry.
They don't really have a choice to "identify" as a jew or not, because there are so few of them. There around 15m jews total (7m in North America, 5m in Israel, the rest scattered) Out of a population of over 7 billion, they're .21% of the world population. I know a lot of agnostic Jews, yet they still identify themselves as Jewish. If you ask a Jew "what he is", he won't identify himself as Polish or Russian or whatever, they say Jewish. If you go up to most white people in America and ask them "what they are", they won't identify themselves as Christian. Christians don't have their own language or homeland. Judaism is also extremely exclusive compared to Christianity.
Claiming they hail from many countries isn't a valid argument either. You can probably find a Satanist in every country, but like Jews, they are a minority. Judaism is more than just a religion, its a way of life; a culture that unites a group of people. It's like being Amish. Amish is an anabaptist christian religion, but I think we all know there's more to being Amish then just worship. Their own language, culture etc.