I am utterly uninterested in anything the NT or the Quran have to say about a Messiah, since the Christian and Muslim concepts of a messiah are utterly foreign to the Tanakh teaching.
The early Christians followed the Tanakh and didn't just follow Judaism but they also believed in the Christian concept of the Messiah. Their interpretations of the Tanakh were different from the interpretations of the Jewish people who didn't believe in Christ. Jewish Christian - Wikipedia
Jewish sects
During the early first century CE there were many competing Jewish sects in the Holy Land, and those that became Rabbinic Judaism and Proto-orthodox Christianity were but two of these. There were Pharisees, Sadducees, and Zealots, but also other less influential sects, including the Essenes.[ The first century BCE and first century CE saw a growing number of charismatic religious leaders contributing to what would become the Mishnah of Rabbinic Judaism; the ministry of Jesus would lead to the emergence of the first Jewish Christian community.
Although the gospels contain strong condemnations of the Pharisees, Paul the Apostle claimed to have been a Pharisee, and there is a clear influence of Hillel's interpretation of the Torah in the Gospel-sayings.[Belief in the resurrection of the dead in the messianic age was a core Pharisaic doctrine.
Messiah in Judaism - Wikipedia
Jewish messianism gave birth to Christianity, which started as a Second Temple Period messianic Jewish sect.