My wife is Messianic Jewish. I presume they are more liberal in their approach than most Jewish sects, but as they all seem to use their phones during Sabbath (even posting videos from services to Facebook)
On the one hand, I don't really want to criticize other people's religion. On the other hand, by all general criteria, your wife is at least as much a Christian as a Mormon or a Jehovah's Witness is, and very few non-adherent Jews would describe her as an observant Jew.
Messianic Jewish are generally not considered Jewish by the community, as their beliefs violate basic tenets of the religion. There is a lot of evidence towards these sects being used to manipulate non-religious Jews into conversion to Christianity. They are still welcome, but considered Christians.
https://en.wikipedia.org/wiki/Messianic_Judaism for people who are interested in a summary.