Here’s the website from the Messianic Jewish Alliance of America, which was founded in 1915.
mjaa.org/messianic-movement/ and they discuss this question.
(There’s also a ‘Messianic Jewish Alliance of Israel.’)
I was involved with Messianic Judaism for many years. Each congregation of Messianic Jews would be different from another, usually depending on the background of the leader. One congregation I went to, the leader was a Rabbi from an Orthodox background. When he came to believe in Y’shua as the Messiah, he remained an ordained Rabbi, but started a Messianic Congregation. It was neither Evangelical, nor patterned after a Christian Church, but was just like attending a synagogue–except for the belief in Y’shua.
Other leaders are from various backgrounds in Judaism and became ordained through a Christian denomination and set up congregations as varied as their backgrounds. I would assume though, that as the movement grows, it may become recognized within itself and set up an ordination if its own through its own educational programs and requirements, but that’s just my thought. .
The people who attend Messianic congregations are a mixture of those from a Christian background and people from a Jewish background who believe the Messiah has already come. Some of these would refer to themselves as Christians, others would not. It would depend on the individual. and the answer would be as varied as the individual and their own background. But the website I mentioned above explains it from a Messianic Jewish perspective.