I actually have a problem with this, even if it is a quote from the HM. The Nazis did not just leave Jews and Jewish identity to be defined solely based on religion. The Nazis saw the Jews (and any others they characterized as Untermensch) as biologically inferior to them, and that is why eugenics played such a huge role in Nazi (pseudo) sciences. If the Nazis merely left Jewish identity to be dictated by their religion, they would never persecute those who converted to Christianity, and many Jews feigned themselves as Christians as a means to survive. An example of this would be the great German philosopher Edmund Husserl, the father of phenomenology; though Jewish, he converted to Protestant Christianity (and legitimately, not for the sake of hiding) but still fled Germany, as he knew (like many others) that in the eyes of the Nazis, once a Jew, always a Jew.

The persecution of Jewish people based on their religion is expansive, going all the way back to medieval times and before (just read about Protestant reformer Martin Luther).

But starting in the mid-late 1800s and especially early 1900s, anti-Semitism and racism in general started to be constructed on a more biological theme.