No one is racially and culturally pure. These are nationalist myths. A quick test of your DNA and comparison to ancient samples (as opposed to modern which obviously will be more homogenous due to proximity in time) will confirm Jews have mixed in the last two thousand years just like Greeks have. Given we don’t live that far apart and have been interacting for thousands of years it’s entirely possible we have some shared ancestry.
There are two main schools of thought in academia when it comes to nationalism today…. modernism and primordialism . The modernist school argues all nationalities are social constructions. Under this worldview both you and I are not “real” Greeks or “real” Jews. We have just been conditioned by our parents and peers to have this perception of ourselves (much like a religion). The second school is primordialism. This argues that nations are repetitions of key themes. (language, and/or religion, and/or territorial association, and/or education, and/or cultural memes, and so on). Only in this latter view we only would be :real Greeks and Jews. My experience most haven’t put much thought into objectively and honestly analysing their identity or that of others. What they tend to do is narrate based on what they see as their interests. A prime example of this is the former Yugoslavians and their foreign apologists… They claim we not “real” Greeks… then ignore the former yugoslavians are Slavs that now claim to be apparently antihellenic founders of the Hellenistic period. (which defies basic common sense.
As for the Palis, DNA testing suggests they are at least partiallly related to ancient Levant people too but it’s not in a direct way. It’s essentially like the former Yugoslavians. They adopted a name in the late 19th and early 20t century but have no cultural, religious or linguistic paper trail to the ancient Philistene or Canaanites. (