The TL;DR of this is that Christians inherited the title of "Israel" or "God's people" after the coming of Jesus and that the true "Israelis" are the descendants of Christians.

This is because Christians believe that Jesus fulfilled thousands of years of prophecies, contained in the Old Testament, predicting a "Messiah." Those who followed Jesus afterwards are the true people of Israel. That's mainly why those "Jewish" books are included in the Christian bible.