It depends on the situation. Today I believe that Messianic Jews should be referred to as Christians. But let's explore the issue.
1. When Constantine was in power, he executed and hunted down people that he said were Jews. But here's the issue. Many of these "Jews" believed that Jesus was the Messiah. But they refused to observe Easter and incorporate Pagan customs into their beliefs. So they were hunted. Up to 500 A.D. there were "Jews" hiding from the Roman empire and/or other Christian nations. But many of these Jews believed that Jesus was the Messiah. If they were Christians, why did the Roman empire hunt them? If you didn't observe Easter, Sunday worship, and observed Passover and the Festival of Booths (Tabernacles), you were executed as a Jew. So Messianic Jews would be considered Jews by Constantine and the Roman empire.
2. Messianic Jews are a scattered bunch. Sometimes even Jews mistaken Messianic Jews for the standard Jew. But some sects are quite disorganized ( I won't name churches). Some are sects consisting of Christians who adopted Jewish customs. Some are Jewish sects adopting Christian customs. I used to work at a world reknowned resort. We had religious groups from all over the world stay at our hotel. There were tons of different Messianic Jew churches, some quite similar, some not.
3. I'm someone who calls himself Christian by definition, but I don't know what I am. I'm NOT a Messianic Jew however. I do observe Jewish customs and follow Jesus. The apostles and Jesus observed Passover and other mosaic observances. The apostles continued to do so after Jesus. So in a way, the apostles could be called Messianic Jews. But they never called themselves that. Matter of fact, they didn't come up with the word "Christian" either. They considered themselves Jews and thought nothing of it at first.