There are several definitions of who is Jewish. There is the religious definition, the ethnic definition and the question of identity. Most people don't identify themselves as Jewish unless they also have one of the other two, but it is possible for a person to be born to converts to Judaism, and therefore identify oneself as Jewish, but not see it as one's religion. It is neither one's ethnicity, because one's parents are not ethnic Jews.
Now, we have a group of ethnic Jews who have turned to Christianism, but who haven't given away their Jewish identity. These people are called "Jews for Jesus", "Hebrew Christians" or "Messianic Jews", which names are VERY misleading. A Messianic Jew is actually a religious Jew who believes in Jewish Messianism, which is something very different from Christianity. It is possible for a Jew to believe Jesus - or Yeshua ha-Notzri - is the Jewish messiah, and still remain Jewish, but it is more likely that one has left the Jewish religion behing oneself and turned Christian. Interestingly enough, the Christians of Jewish ethnicity don't approve these true Messianic Jews as "Messianic Jews".
These are the important questions that make all the difference:
Was Jesus a man or God?
If you think Jesus was in any way or form divine, God's son more than any other human being, part of the thrinity, then you are not Jewish, but Christian.
What is the significance of the Greek Scriptures?
Is it of the same importance as Tanakh? If so, then you are not Jewish, but Christian. One must understand here that what is said in the Greek Scriptures - what they claim Jesus taught - is against Torah, and if you believe Torah is God's Law, given to your people by God Himself, you do not think scriptures where you are told the God's Law is wrong, are of any worth.
If you think the Greek Scriptures are right and Talmud is wrong, you are not Jewish. What is taught in the Greek Scriptures goes againt TORAH, what is taught in Talmud, does not.
This is not a question of interpretation and opinion, but theology and dogmas.
Just like there cannot be Pagan Christians, there cannot neither be Jewish Christians, Muslim Christians, Buddhist Christians or any other such Chimera. Considering how highly the Christians speak about their religion, it's surprising how much they want other people's religions too...
And why am I writing this? It has nothing to do with Paganism...
It's because my Jewish husband is being persecuted by missioners at YouTube, and I hate missioners, especially missioners, who will not accept that I have already found God and I am very pleased with Her and would not change Her to anything in the world, especially not to something like Jesus. That's like asking me to exchange my woolen clothes in a blizzard to a wet polyester satin sheet.