Were Jesus himself and Paul and Peter and the other disciples not Jews because they believed Christ was the Messiah? Did Christianity not essentially start as a Jewish sect?
Today, and arguably even back then, Judaism is more defined as an ethnicity than as a religion. If secular Jews who are essentially atheists can be Jews, why can't those who profess to believe Christ was the Messiah?
After all, professing to have converted to Christianity did not save Jews from the Holocaust. Ultimately, it's clear that Jewish identity is something more than believing in certain religious ideas and certainly much more than just rejecting Christ.
There is a very good YouTube channel called Useful Charts that has covered religious groups/denominations and their family trees before, and he is a Jewish convert (i.e., not ethnically Jewish), but his wife is both ethnically and religiously Jewish. He described this in a way that I think is very articulate, and I agree with it:
Judaism is still essentially a tribe at its core. And whatever you want to say about the dynamics of Late Antiquity, it is pretty undeniable that over the last 2,000 years the "Jewish identity" has evolved to have a meaning that more or less necessitates NOT being Christian. If you are Christian, you have - in the eyes of 99% of Jews - "left the tribe."
This doesn't necessarily correlate with how "open" the person's preferred current of Judaism is otherwise, either. For example, an Orthodox posek would probably consider a Jewish convert to Christianity a heretical and idolatrous Jew but still a Jew, whereas Reform Judaism (forces within which formerly aspired to overhaul Judaism into something that was "just" a religion without any ethnic or national components) customarily rules anyone who converts to Christianity out of the category "Jew" entirely.
There's also a difference between a Jew who happens to convert to Christianity and "Messianic Judaism" in the strict sense, which Sanchez touched on. The term is most often used to mean a specific theological and liturgical style that retains elements of Jewish practice in what most people, especially most Jewish people, feel are unsettlingly recontextualized and revaluated ways; this is what he was arguing with the other Bob Dylan fans about on Facebook.
Cardinal Lustiger was not a Messianic Jew.