There may be a couple of issues involved here. While the intellectual association of Jesus with the Jews and the Jewish religion has never been lost, there has long been a dissociation between Christians and the Jews and the Jewish religion.
It should be understood that Judea was a Roman province in Jesus' time. The Romans were interested in seeing how their gods appeared in the religions of other lands, but they obviously found no syncretism in Judaism, a religion that proclaimed One True God and also declared all others false gods. Nonetheless, the Romans tolerated it in part because Judaism was limited to the Jews and the Jews weren't too diligent about making proselytes.
Enter Christianity. At first, Christians were incredibly similar to Jews. The religion was practically identical. The first leaders of Christianity were all Jews: Jesus, his brothers, the Twelve, Paul, even Timothy. The Christians made themselves unwelcome in a couple of ways: they actively sought converts and they forbid those converts from worshiping the Roman gods. The Romans believed that their piety and the public nature of their religion was responsible for the strength of their Empire and this new, private religion was destroying that.
The Christians suffered persecution as a result. They suffered further persecution when the Jews rebelled against the Romans (A.D. 66-135) because they were still fairly closely tied to the Jews. Eventually, the Christians began dissociating themselves from the Jews, possibly because that association was detrimental, possibly because they had started incorporating elements from the religions surrounding them, possibly because God was Two or Three Persons while the Jews continued to emphasize that God is One.
By the fourth century, Christians were eager to break their ties to the Jewish faith. At the First Council of Nicea, they separated calculation of Easter from the Jewish month of Nisan (though the revised calculation has almost identical results). Shortly afterward, the Council of Laodicea outlawed resting on the Sabbath (like the Jews) and encouraged resting on Sunday instead. The Council of Laodicea also forbid several forms of socialization with the Jews.
The Renaissance comes another millennium after all this. Most Christians do not have a copy of the Bible for themselves and few could read one even if they saw it. Christianity has completely separated itself from Judaism to the point that the Jews are now enemies to the Christians. The Jews were the scapegoats for many of the woes that befell the Christian nations. They accused the Jews when the Black Death killed millions of Christians but apparently left the Jews alone. They accused the Jews of killing Jesus.
This anti-Semitism is apparent in many translations of the Bible today. The New Testament is rife with Jewish names, but you might have trouble seeing that from the English translations of them. Of the six Maccabees, five (Mattityahu, Judah, Yochanan, Eleazar, and Simeon) show up in the gospels (Matthew, Judas, John, Lazarus, and Simon). Several other Jewish names appear, too, such as Jacob (James), Miriam (Mary), Elisheva (Elizabeth), and Joshua (Jesus).
After going to such lengths to dissociate from the Jews for over 1,000 years, it is understandable that the typical Christian was not consciously aware that Jesus was a Jew. They called Him Christ, as though it were Jesus' last name, unaware that it was the Greek translation of Mashiach. Mashiach, "anointed one", was originally used for priests, but was only used for kings after Saul's day, so it became synonymous with "king" for the Jews.
Why didn't they "discover" that Jesus was a Jew until the sixteenth century?
- Christians were enemies of the Jews.
- The Jews didn't believe in Jesus.
- Martin Luther didn't encourage people to read the Bible for themselves and ignite the Reformation and Protestant movement until the 16th century.