I think that the lyrics and or the intent of the song is what makes it Jewish.
Take for example Norman Greenbaum’s 1969 hit song, “Spirit in the Sky”. Greenbaum is an Orthodox Jew and if you listen to the lyrics it is very much not a Jewish song. However, Greenbaum’s intent was to express spiritualism and used Christian overtones to express it because it was more marektable.
If you were to take the song at the lyrics it is not a Jewish one. But if you take the intent of the song behind the lyrics knowing that Greenbaum is an Orthodox Jew, you can make a compelling argument that it is a Jewish song. To quote Greenbaum (i got this off his Wikipedia page),
Sometimes its hard to define what “Jewish” music is.