He means that his name is both American and Dominican. His name speaks to the two worlds he lives in- - the world of a normal American kid and his cultural world of Spanish legacy. They propose two unique societies - Jonathan is more European and Rodriguez is Spanish. He was brought up in New York City, and furthermore rural Connecticut.
This intermingling of powers was a determining condition in the struggle between church and state that was a main theme in the history of the West in the Middle Ages