The notion of a multicultural society where you can choose to check a box with the belief system with which you identify is ... a newer one. (In the Soviet Union, your passport could call you a "Russian", a "Ukrainian", a "Georgian" ... or just a "Jew." Didn't matter what you worshiped or believed.)
In older times you would identify a person as a "Hebrew", or later an "Israelite", and their religion was kind of a given -- whether they kept a lot of it or not. Or you would mention the name of the God you worship.
Moses tells Pharaoh he is speaking in the name of "God, the Lord of the Hebrews." Later, Jonah tells the sailors that he is "a Hebrew, who fears the God of the Heavens." (In both cases, the specific word used for "God" probably also helped with identification.)
When the North Kingdom of "Israel" is defeated and all that remains is the South Kingdom of "Judah", and then the South Kingdom is defeated and taken into exile, they become known as "Judeans" or "Yehudim." The Book of Esther, trying not to offend the sensibilities of the Persian Empire under which it was written, sidesteps "religion per se"; it simply says that documents were written "to every nation and people by their alphabet and language; plus to the Judeans in their alphabet and language." (Similarly it doesn't say the Jews "prayed", just "mourned.")
In the Mishna, a Jew is called a "Yisrael." A Jew who has taken up another faith is Yisrael she-hemir dato. "An Israelite who downgraded his dogma." It's understood from context that the natural "dogma" of the Israelite was what we call Judaism.