No. Messianic Jews have not converted to Christianity, but have accepted the blood atonement of their Jewish Messiah as prophesied in the Scriptures.
Although it is correct that both Christianity and Messianic Judaism focus on the Jewish Messiah, Yeshua, there are very distinct differences between Messianic Jews and Christians, primarily in the ways we express our faith. We, as Messianic Jews continue to practice Jewish traditions, feasts, and other festivals as instructed in the Torah. Christianity, on the other hand, celebrates more modern interpretations of the traditional festivals, such as the Jewish Passover and the Christian Easter.
The important thing to note, however, are the similarities between Christians and Messianic Jews. According to 1 Corinthians 12:12, all the parts together form one body under the Messiah.