What made the Mediterranean unique? Geography, simply nothing like it on earth honestly?
It isn't and wasn't unique. Of course, it's not exactly a common thing: a large sea, separated from the larger ocean, with lots of potential for (in various ages) having multiple coastal cultures interact,
and having one empire control the area (as Rome did in OTL, by eliminating the competitors and making it their "mare nostrum"). An open ocean doesn't have a coast on the other side, meaning you'll always be travelling along (or rather: parallel to) the coast to some extent. For instance, the indian ocean trade network. That's pretty basic east-west interaction, along a coastline to the north. There isn't an entire cultural sphere surrounding the sea-area in question.
But does that make the Med unique? Not really. As
@Soverihn points out, the Caribbean was such a cultural sphere, at least to an extent. While not as closed off from the open ocean, I don't think that's what defined the Med. I think it's the fact that a relatively easy-to-cross sea was surrounded by coasts on all sides. That's what made it so likely to be an interaction sphere, and potentially, a sea that can become entirely surrounded by one empire.
But why didn't the Persian Gulf, Caspian Sea, the Great Lakes in North America, or the Gulf of Mexico develop in parallel?
Because cultures don't develop in the same way, at the same time. They don't have the same means, the same priorities etc.
Environment does shape a culture, certainly. But even if you took the Med, changed nothing about it, and simply put completely different cultures along its coasts, those cultures may not act as the OTL cultures of the region did. I suspect they'll still tend to trade oversea, but their cultural preferences, their tech level, and the extent to which they are tied up in overland affairs... that all plsys a role. (For instance, a culture bordering the the Med but almost exlusively preoccupied fighting a rival power further inland is going to be "with its back to the sea", and may simply have neither the time nor the inclination to get involved in the whole seafaring business.)
With a POD as far back as you want without ASB, is it possible to have any other body of water have a history similar to that of the Mediterranean of being a unified continium of cultures, empires, etc.
The best shot for approximating the conditions you describe as pertaining to the Med would, in my opinion, be an ATL focused on the Caribbean. Mostly because it already had the basic setup for a cultural interaction sphere, and has mostly surrounding coastlines, thus making it most like the Med in those regards. Also: a history of city-states in Mesoamerica, and the Maya did in fact carry out Caribbean trade. We discussed this quite recently, coming up with a bit of an ATL where the Maya develop an Caribbean trade "empire". The discussion thread for that is
here. You may want to check that out: the general scenario we eventually came up with seems to be exactly what you are looking for.