He is not Jewish if he believes Jesus is the Messiah......See the following criteria. According to the website My Jewish Learning, they explain as follows:
"The
Prophets (Nevi’im), who wrote hundreds of years before Jesus’ birth, envisioned a messianic age as a period of universal peace, in which war and hunger are eradicated, and humanity accepts God’s sovereignty. By the first century, the view developed that the messianic age would witness a general resurrection of the dead, the in-gathering of all the Jews, including the
10 lost tribes, to the land of Israel, a final judgment and universal peace."
We know through Jesus' ministries on earth world hunger still existed and there was no world peace based on the above description. Furthermore:
"
Jews for Jesus is one branch of a wider movement called Messianic Jews. Members of this movement are not accepted as Jewish by the broader Jewish community, even though some adherents may have been born Jewish and their ritual life includes Jewish practices. While an individual Jew could accept Jesus as the messiah and technically remain Jewish — rejection of any core Jewish belief or practice does not negate one’s Jewishness — the beliefs of messianic Jews are theologically incompatible with Judaism."
The above addresses your reference of a Jew believing Jesus as the Messiah, something incompatible with Judaism.
In the following, Maimonides addresses the issue:
"Maimonides, in his
Mishneh Torah, describes Jesus as the failed messiah foreseen by the prophet Daniel. Rather than redeeming Israel, Maimonides writes, Jesus caused Jews to be killed and exiled, changed the Torah and led the world to worship a false God."
See:
What Do Jews Believe About Jesus? | My Jewish Learning
Christians are better off siding with Muslims on the subject of Jesus than Jews IMHO.