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Consider, for example, an Irish Catholic. If he converts to another religion, he's no longer Catholic. But he's still Irish. Everyone can understand this, because there are distinct words for the national/ethnic identity and the religious identity.
But the word "Jewish" can refer to either one, and that's what's tripping people up.
This is a great example for how confusing this all becomes in modernity.
What were the Irish people? Various tribes who lived in Ireland, identified by some combination of locale and way of life (not an Irishman if you were a Celt from Gaul, nor if you lived in Ireland as a Germanic). Way of life was not arbitrary - it was culturally and linguistically distinct, with a kind of shared mythology and national identity that was expressed through religious and governmental practice. In ancient times this wasn't their DNA. Ancients didn't know what DNA was.
If you were a celt from not Ireland you were not Irish. If you were a non-Celt in Ireland you were not Irish.
Then those people converted to Christianity. They became something different. They were no longer what they were, and there was a distinction made between pagans and Christians as different peoples. This is how the ancients understood it, this is why Christianity was so disruptive - why it was so infuriating for the pagans when the Judaeans or Christians refused to sacrifice to the gods. They were not being part of the people.
The only way your statement "if he converts he's no longer catholic" begs the distinction between the people group and the religion - which is to say, the entire point. In ancient times, if you stopped doing the cultural thing of your people, you were no longer part of that people. If you stopped doing the religious festivals of Athenians you were no longer an Athenian. If you stopped worshipping Marduk you couldn't be Babylonian. You've become something else. The right statement is "if he converts he's no longer Irish".
So let's then explore what a Jew was. There are no "Jews" in the Torah - there were Israelites, some of whom were of Judah. Then there was a civil war and the people of Judah became distinct from the people of the northern kingdom of Israel. How? Tribal affiliation, which included in its essence religious rites and festivals. Then later there was the exile and return. How did Jews in Greek-dominated Judaea distinguish themselves? Way of life - circumcision, dietary restrictions, way of worship, and sure language and dress. The Greeks called this the nomos, the same word they used to translate the Hebrew torah. At the time of the second temple then, you were a Jew because you were one of the two- three-tribes who lived in Judah, who practiced the way of life of the people of Judaea, which included several strands of related religious sects (all who claimed the others were in some way deficient or false). "Jew" just means "Judaean" the same way "Irish" means "Irishman". You could also be a Judaean by maintaining the way of life of the Judaean outside of Judea, like our German living in Ireland. In the diaspora Jews did this.
So - be a Judaean you needed to be from Judaea and practicing the way of life of a Judaean, or living the Judaean way of life outside of Judaea as an identifier to continue your identity as distinct from everyone else. We all know Judaea didn't exist as a place for identity of a people group for a long time, so that leaves only the Torah, the common way of life.
The Judaean way of life is something of historical fact: religious belief in Yahweh and following the Torah. Nobody in the second temple era would have accepted that you were a Jew if you weren't circumcised, if you didn't keep the Torah (i.e., the Sabbath), if you worshipped foreign gods. If you told someone you were an atheist Jew, meaning you didn't believe Yahweh existed, that would have been sheer nonsense to them. Or as one second-temple era Jew put it - "not all who are of Israel are Israel" or as another noted Jews of his day said "neither will we transgress the commands of our law; and as we depend upon the excellency of our laws, and, by the labors of our ancestors, have continued hitherto without suffering them to be transgressed, we dare not by any means suffer ourselves to be so timorous as to transgress those laws out of the fear of death, which God hath determined are for our advantage...we will die before we see our laws transgressed."
Anyway all of this gets really challenged when you say: can you be a Jewish Christian? If your answer is "yes" then on what basis is one a Jew? maternal DNA? If your answer is "no" then on what basis is one no longer a Jew? merely being non-Christian?