Jesus was a Jew teaching his followers. I have heard Jews refer to him as a Rabbi.
He said that he didn't come to destroy the (Mosaic) Law, but to fulfill it. (Matt 5:17-20)
Therefore, are Christians allowed to call ourselves Jews?
I think - Jews are Jews [exclusively for Israelites]
During the time of the prophet Jeremiah, God promised a new covenant.
“The days are coming,” declares the Lord, “when I will make a
new covenant with the people of Israel and with the people of Judah.
Jeremiah 31:31
My Lord Jesus was called Rabbi because he lived under the Law:
But when
the right time came, God sent his Son, who was
born from a woman and lived
under the law.
Galatians 4:4
It is true that my Lord Jesus, as you quote it, didn't come to destroy the (Mosaic) Law, but to fulfill it. (Matt 5:17-20) - but his mission was to seal God's new covenant which was promised.
And in doing so, he built his church (Matt 16:18) that is when his followers stop being Jews and started being Christians.
So I tell you, you are Peter. And I will build my
church on this rock. The power of death will not be able to defeat my
church.
Matthew 16:18
The new covenant is in the blood of my Lord Jesus Christ:
In the same way, after the supper he took the cup, saying, “This cup is the
new covenant in my blood, which is poured out for you.
Luke 22:20
The disciples are not called Jews after Christ died on the cross and after God raised him from the dead. In fact they were called Christians.
and when he found him, he brought him to Antioch. So for a whole year Barnabas and Saul met with the church and taught great numbers of people.
The disciples were called Christians first at Antioch. Acts 11:26
The Jews in fact, persecuted the first Christians and Saul (who became apostle Paul) testifies to this:
Galatians 1:13 New Life Version (NLV)
You have heard of my old life
when I followed the Jewish religion. I made it as hard as I could for the Christians and did everything I could
to destroy the Christian church.