Jewish Soul

In the play, I speak about the concept of a "Jewish Soul," the idea that someone is born with a Jewish soul even thought they're not born into a Jewish family. I asked Rabbi Aaron Flanzraich at Beth Sholom synagogue in Toronto about this, and his view was that a Jewish “soul” is the by product of being born as a Jew, much as a Christian soul would reside in a Christian.

However, in places like here, here and here (among other places) I read that a Jewish soul can in fact be born into a non-Jewish person.

Is this more of a Kabbalistic principle than a mainstream Jewish one? Or is it in fact more open to interpretation and debate than that? What's your take?