Not true. Christians kept the Old Testament as part of the Holy Scriptures. And the Christian religion came from the Jews. First Christians were Jews who believed that Messiah came. Non Jews became Christians later.
Not so with Islam. Islam was founded outside of Church and Synagogue, was not based on Jewish or Christian writings - instead made false claims about them.
From Constantine through Jews who refused to accept Jesus were frequently considered to be in open defiance of Christ for refusing to accept him as Messiah like the early Christians did.