Despite what US law holds - and maybe laws in other countries for all I know - there's no such thing as a Jewish race. Race denotes shared ancestry and shared genetics. And while you are thinking, "But they do share an ancestry and genetics!" most of us do, but some of us do not. Why, you ask? Because of *points at converts" those guys.
You see, anyone can convert via Judaism and thus be recognized as Jewish/a Jew. However, as an example, no matter how much I like rap music, I cannot convert into the black race to become a legitimate rapper.
See what I'm say'n? Yo.