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The Greeks saw themselves as surrounded by people who worshiped the same gods under different names because they were literally surrounded by people who worshiped the same gods under different names!
Greek is a Proto-Indo-European (PIE) language, which is to say it can be traced to the people who first domesticated the horse North of the Black Sea in roughly 3,500 BC.
It turns out that that basically all of Europe (including notably Germany, Italy, and the Balkans) as well as Anatolia (Troy, Hittites, etc) and Persia (Parthia) also spoke PIE languages. It stands to reason that the PIE religion spread with the language.
The Semantic speakers of the Levant, as well as the Egyptians and Sumerians all came from different language families and obviously would not have inherited PIE myths, but they may have been influenced by them. The presence of the Flood Myth in early Vedic and Greek sources suggests that it may be PIE in origin, would would imply a PIE influence on both the myths of the Levant (Biblical Flood) and Sumer (Gilgamesh Flood).
It may be that the Levant and Mesopotamian cultures absorbed enough of PIE myths from Anatolia, Persia, and Greece into their own religion that they became recognizable - especially if the Greeks were already accustomed to finding their "own" gods in other cultures.