I'm about to do something I said would not be tolerated here. If someone wants to boot me...
I made a policy of not discussing other religions. But I see that there is much misunderstanding, and much resentment on this topic.
First of all, what is a Jew? Our Tradition recognizes one as a Jew who was either born into that Nation that made a Covenant with G-d at Sinai, or who had duly joined that Covenant by means of conversion. "Feeling Jewish" is not a criterion, any more than my deciding that I like your family, and am therefore your brother.
Once Jewish, one cannot be made non-Jewish, even if they should sacrifice to Zeus every day. They are, however, violating the covenant. A covenant is a contract. One party not keeping the terms of a contract does not mean that the Contract is null and void. They will, however, face consequences. I have a close friend (of 40 years standing) who is a Wiccan Priestess. She worships many gods and goddesses. But she is born Jewish, and darn proud of that. If you tell her she is not Jewish, you had better duck...fast! She has two children, who are, according to halachah, Jewish. No Orthodox rabbi would ever say differently.
Now, we must talk about Jesus. Some question if he even existed. There is only one contemporary mention of him, and that is likely a forgery. He could very well be a composite figure. Many of the events, in both Jewish and Christian sources, do not jive chronologically. The name "Yeshua" was the second most common male Jewish name at that time, after Simeon. Confusion between figures is certainly possible. I will go here by the assumption that the stories in the Gospels are substantially historical, albeit written from a non-objective stance. Let's start at the beginning. Mary was "Betrothed" to Joseph, who had not yet "taken her". See my series about Jewish marriage. That means that she had had "kiddushin", the part of the ceremony when she becomes forbidden to any other man, but is not yet permitted to her husband, but not "Nissuin", which permits her to him. We know from Jewish sources that the custom in Judea was to separate these ceremonies by 12 months. (Today, we do both together, separated only by the reading of the Ketubah). According to Torah law, that made Jesus a "Mamzer", a bastard born of adultery. It would invalidate him from ever marrying, or holding any official position. He would be an outcast, a pariah. (I do not wish to discuss now the morality of this rule). Now, when we look at what Jesus is alleged to have taught, we see a great deal of frustration and anger against what he perceives to be injustice. This could very well be a reflection on his personal situation. Beyond that, about 90% of what he said is completely consistent with Jewish law and tradition. In many ways, he can be seen as a continuation of Hillel, who lived a centurt earlier, who put Man and human dignity first. However, there is nothing new in these teachings, only a different emphasis. As to the other 10%, there are denials of basic Jewish concepts (It has been said unto you...but I say...), violation of Shabbat in non life threatening situations, and at least broad hints of both Messiahship and Divinity. (Although these statements can be otherwise interpreted). There is little doubt that he would have been seen as a heretic, although there are contradicting stories in the Gospels about his relationship with the rabbis. The upper classes (the ruling class and the Priesthood) were vassals of Rome, and feared any talk of a King of the Jews. (Crucified criminals had a sign over their cross indicating their crime. Hence, the sign "King of the Jews" over Jesus' cross.) His followers were common people and the disenfranchised. There is no indication that he had a following among the learned or devout. In terms of the Jews, he is a footnote on one page of Jewish history. As a Messiah, he had failed miserably. He was just another pretender to that role, one of scores. Then comes Paul. Paul never met Jesus in the flesh; only in vision. Paul rarely quotes Jesus, or makes reference to the events in his life. Paul's message is overwhelmingly about Jesus' death. Paul is a Hellenize Jew. The Greeks believed that the gods sometimes came to Earth, born as humans. (You will recognize this idea from its Hindu version; the Avatar). This form of the gods is called "Soter" in Greek, meaning Savior. Paul made a Jewish heretic into a Greek god!. Among early Christians, there were both Jews and non-Jews. Paul posited that all believers were grafted on to the original covenant (Romans 9 and 11), which no longer requires observance of the laws "If there is salvation under the law, Christ died in vain" (Galatians 2:21). (Salvation itself was a Pauline concept, but let's leave that for anther day). The Jews were henceforth flooded with non-Jews, claiming to be Jews "grafted by faith". The rabbis saw this as a grave danger to Jewish identity, and composed an anti-Christian "blessing" that was inserted into the amidah prayer. (For the last thousand years, it has been changed to a curse upon "evil doers and slanderers".) This forced Christianity to go its own way. Nearly all Christianity after this point is antisemitic. Fortunately, the most virulent groups were thrown out at the Council of Nicaea, in CE, albeit not becasue of their antisemitism. Persecution of Jews continued from that time until today, in various guises (NOT only Catholics). More Jews have been killed in Jesus' name than any other cause. It is no wonder that for most Jews, the name "Jesus" is emotionally on a par with Hitler. Christian missionaries had very little success with Jews, until "Moshe Rosen" (ordained and funded by the Southern Baptist Church) founded Jews for Jesus in the 1960s, which presented the option of being Jewish and following Jesus. Before the 1967 Six day War, most Evangelicals were anti-Israel, but switched to a supportive position as it seemed that Israel was fulfilling the prophecies in the Book of Revelation. Thus was born the "Messianic" movement.
Jews today take different stances concerning Jesus. Some see him as an evil, almost demonic, figure. Medieval legends and superstitions have been melded into a kind of paranoia. At one time, Jews in Christian Europe were not allowed out in the streets on Christmas Eve, so as not to pollute the "Holy Night". Christmas, in Italian, is Natale. Jews could not go to synagogue or study halls. Today, many Ashkenazi Jews do not study Torah on the first half of Christmas Eve. The reason has been forgotten. The belief has become that if we study, we are giving Jesus "energy" (chiyut). I know many who are "strict" to avoid study on Roman, Greek and Russian Christmases! Others, including yours truly, see this as ludicrous. On the other hand, most Reform, and many Conservative rabbis, are perfectly willing to give Jesus "billing" as a Jewish thinker, minus Messiahship, Divinity...or Paul. For many others (including me), he is simply irrelevant, an "also ran". But, as I have shown elsewhere, most rabbis regard Christianity as acceptable for non-Jews. As such, it has brought them to G-d, and we await their coming closer. I freely and sincerely wish my Christian friends a "Merry Christmas", as they greet me for my festivals. (I usually get unfriended by a few people every year for this). So, I have little regard for Jesus, or for Christianity. We have suffered immensely becasue of him. But over a billion people acknowledge the G-d of Israel through that faith. I see it, as RAMBAM states, as a necessary stage in the development of Man's quest for G-d. I look forward to human progress in that area.