Maybe there's a bunch of different strains of "Jewish Christianity" ? When I was growing up, my folks got us all involved in a group like this -- but we called it Messianic Judaism. It was mostly Christians who were dissatisfied with the smorgasbord of Protestant options and wanted to "do it like Jesus did." In other words, a bunch of Christians who worshiped like Jews. We'd come together on Saturday nights and have a havdalah service, sing, dance, and learn the Bible (both OT and NT). I actually look back on those times fondly, and still try to make challah whenever I can.
It ultimately brought me to Orthodoxy, so I can't complain.
Jewish Christianity is the religious outlook & perspective adopted by a Jew
In my journey to Holy Orthodoxy I was involved with "Jews for Jesus" to some extent. And at the time I was a fallen away Roman Catholic who was disillusioned by the post happenings of VATII. Keep in mind that my journey involved from not going to church at all to investigating other forms of Christianity. I looked into Evangelicals, Pentecostalism, Anglicanism, etc. I hit upon Jews for Jesus and thought that maybe this was a fundamental means to salvation only to learn that all they were was a pseudo Protestant group using the OT as a way of luring Jews into Christianity. Once I found Holy Orthodoxy I knew that this Jews for Jesus did not fulfill my need for the ancient and mystical Church of Jesus Christ.