Even as a Christian, I do not believe I have the authority to claim who is or who is not a Christian.
As for Messianic Jews demanding to be recognized as Jews, yes, there is a lot of evidence. They have constantly attempted to take advantage of Israel’s Law of Return and lost all attempts to be recognized by Jews until 2008. While a minor victory was won in their case they were pushing at the time, their battle did not end then. At present some Messianic Jews can be recognized as Jews but are still not entitled to immigrate. The battle is ongoing and you can Google all the various and countless news articles for yourself.
As a Roman Catholic I find it appalling to demand that any people, ethnic group, race, etc., be forced to give up its cultural identity and ethnic customs and practices as a requisite to joining the Church. Catholics do not practice such things today. No people, Jewish, Irish, Mexican, black, white, Asian, Indian, etc., should be forced to assimilate to another culture upon becoming Christian. Which one should they assimilate to? Should Jews be forced to accept Western American culture practiced by middle class Protestant families? Except for the Jews, aren’t all cultures originally from pagan cultures of the past? Of all people why do the Jews have to give up their culture? Mexicans who enter the Church don’t. Irish people don’t. But Jews do? You might believe that, but Catholics don’t.
The concept of Messiah is a Jewish one. The Old Testament is a product of the Jewish religion. Except for Luke, the entire New Testament was composed by Jews. Have you never read Acts 21.15-26? Are you unaware that after Paul wrote most of his epistles to the Gemtiles that he still claimed to be a Jew, worshipped at the Temple, and did not consider any of his teachings to be applicable to Jewish Christians who wished to remain Torah-observant? Scripture in Acts 21 show the Jewish Christians of the original Church were zealous for the Law and Jewish customs.
And as for your final point, yes Judaism is beginning to recognize some Christians as Jews. As mentioned before, I have a friend who is Catholic of Sephardic Jewish ancestry. The Law of Return to Spain is recognizing descendants of Jews persecuted during the Spanish Inquisition and granting them citizenship, with the Federation of Jewish Communities in Spain recognizing these people as Jewish. My friend, who is Catholic, is one of these people. And no, he does not have to leave Catholicism or practice Judaism to be declared a Sephardic Jew by the Federation in this case.