Definition of Jew 1 : a person belonging to a continuation through descent or conversion of the ancient Jewish people 2 : one whose religion is Judaism 3 a : a member of the tribe of Judah b : israelite 4 : a member of a nation existing in Palestine from the sixth century b.c. to the first century a.d.
Definition of messianic 1 : of or relating to a messiah this messianic kingdom 2 : marked by idealism and an aggressive crusading spirit messianic zeal on a messianic mission
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“Jewish people” are not all that ancient. Judaism came into being in the second temple period, after the Macabee revolt, but they never identified as “Jewish” until the term was created in Europe centuries later.
Most genetic Hebrews are definitely not “Jews.”