But is the recipe the same or different?

Similar. It seems the dish had both Arab and Jewish roots. In colonial Singapore, both communities were small but left an outsized mark.

https://www.peranakanmuseum.nhb.gov.sg/learn/digital-resourc...

The Jewish community in Singapore is mostly Baghdadi (Iraqi Jews who immigrated to British India and later Singapore and Hong Kong). That's why it's the same recipe for Bamia.