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Well of course Judaism is a religion, whereas "Jew" is a race, the descendants of the Tribe of Judah. I don't understand the point to were trying to make.
The reality is that the title of 'Jew' does not... mean a race. Sammy Davis Jr. was a Jew (he converted to Judaism).
The Jewish historian Josephus (100 A.D.) said the title of 'Jew' is the name the small remnant that returned to Jerusalem after the Babylon captivity took for themselves, and comes from the tribal name Judah. He also said all that lived in the land (foreigners included) took that name (and thus their religion).
The only ones who went captive to Babylon for 70 years was the "house of Judah", the three tribes of Judah, Benjamin, and Levi, and the strangers with them. Only a remnant of those returned. The majority of Israel, the ten tribes, had already been removed from the holy lands about 130 years prior to Judah's captivity to Babylon.
Abraham was a Hebrew, not a Jew. Nor were the ten scattered tribes known as Jews. Only the tribes of Judah, Benjamin, Levi, and a small remnant of the ten tribes that left the north when Jeroboam setup the golden calf idols became the Jews.
All of the 12 tribes are Hebrews, their ancestor being Eber, where the name Hebrew is derived.
Later on in history, the nation of Khazaria adopted Judaism as their religion, and they became religious Jews. There are 2 main branches of Jews, the Sephardic which claim heritage from the bloodlines of old Israel from the Middle East, and then the Ashkenazim of Eastern Europe which were mostly converts, like from Khazar. Then there are Ethiopian Jew converts that are of the Black race that have existed for centuries.
So in most ways today, the title of 'Jew' is very much like the word 'Christian', regarding who can be one. Convert to Judaism, and boom, you're a Jew. Convert to faith on Jesus Christ and boom, you're a Christian.
Another reality is that the majority of Israelites NEVER WERE JEWS. The tribes in the north that made up the "kingdom of Israel" were not known as Jews. They were scattered first to the lands of Assyria and the Medes, never to return to the holy lands. They lost knowledge of their heritage as Israelites. They became as Gentiles scattered among the Gentiles, as God had promised He would do to them if they rebelled against Him (Deut.4 & 28). But Judah (Jews) kept their heritage as Israelites.
However, too many deceived get sucked into the lie that false Jews put forth today about heritage from old Israel. Because God has hidden the majority of the seed of Israel today through the scattered lost ten tribes, today's Jews like to get all puffed up and claim sole ownership as God's chosen seed. In reality that is a LIE. And it is even stealing.