I don’t think it makes sense to think of Judaism as a race, and I don’t think anyone should consider it a race, because doing so is often oppressive.
Judaism has its roots in the tribal religion of the ancient Hebrews. Does this mean that any descendants of the Hebrews are part of the Hebrew “race”? By that logic, there’s a Hindu race and a Zoroastrian race that anyone with Aryan ancestry belongs to. Just because a religion starts off as a tribal cult doesn’t make it a race.
Also, the idea of Judaism as a race has more often than not been used as a form of control and oppression. The most obvious example of this was the Nazis, who believed that any members of this mythical “race” of Jews should be wiped out. But anti-Semitism has existed in lots of places, and it has almost always been based on the assumption that “if you are born to Jewish parents, you are part of the Jewish race and are a second-class citizen.”
But Judaism-as-race is also used as a form of control by Jews. I come from a Jewish family, and in their eyes, this makes me a Jew. Nevermind that I am an atheist and I find the religion of Judaism morally despicable. I’m still a Jew, according to some Jews, because I came out of a Jewish vagina, and apparently Judaism is magically passed down matrilinearly through mitochrondrial DNA. This is a form of control. They are saying that I cannot choose to be part of another culture or religion (or lack thereof) or to disassociate myself from my ancestors.
The nation of Israel is a good example of why thinking about Judaism is a “race” is a terrible idea. Israel is an apartheid state where, if you are member of this magical race of chosen people, you are treated as a first class citizen; if you’re not a member of this race, you are a second class citizen. I can become a Jewish citizen easily because of my mom’s magical vagina/mitochondrial DNA; someone supportive of Israel and Jewish culture, on the other hand, has to jump through many hoops to achieve Israeli citizenship.