I hate to break the news to folks, but most “Jewish songs” are in the style of contemporary non-Jewish music, or even actual copies.
Two examples:
1. The most common tune used for “Maoz Tzur” is a German melody that was used by none other than Martin Luther HaRasha for Christian liturgical music.
2. The nigunim used at the oldest continuously operating Jewish congregation in America, Mikveh Israel in Philadelphia, are in the style of the music that was popular in 18th century colonial America.
I dare anyone to find a Jewish composer other than Arnold Schoenberg who wrote music that was not influenced by non-Jewish styles. (And I dare anyone to try to sing any of Schoenberg’s atonal compositions at a simchah or in shul. He did write a setting of “Kol Nidre” but I’ve never heard of anyone trying to use it on Yom Kippur.)