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I often say this, but I do think that we underestimate today how connected the ancient world was (quite possibly in large part because the conflict between Christianity and Islam cut off links between East and West much later). There were commodities like spices traded over huge distances, for instance. In Plato's case, there is a mysterious strand of Egyptian influence in some of his work. It is likely that Egypt was partly so powerful because it was connected both to Europe, what we now call the Middle East, and to the Indian sub-continent. It's all speculation, but I certainly wouldn't assume that Plato could have referred only to 'local history'.
As for the Mediterranean, that is thought to have formed more than 5 million years ago, so there is not likely to be too much human folklore left around that. There certainly will have been many sea level changes, submerging among other things cities like Pavlopetri, and it's likely that that will have generated stories, but again it's hard to connect this to an event like the purported sinking of Atlantis, which Plato explicitly claims was outside the Mediterranean, in a 'real ocean'.
If the Mediterranean and Black Seas were created from sea level rises via the straits of Gibraltar, then low lying cities in the region would have become submerged over time. That’s my view, Plato would have only related to local history not global events.