Joe, there are Jews who identify as Arab Jews, but there are enmormous complexies around doing so, and it's a political statement, rather than a messianic one.
The choice to use the term by non-Jews is often (though not always) used as a form of Mizrahi erasure, such as when Al Jazeera publishes articles/op-eds about the "myth" of the Mizrahi Jew, calling them Arab Jews instead.
Any time someone overrides someone else's self-identification labels, we need to take a look at their motivations.