Okay, so realistically, production and gold are going to be the bottlenecks to keep this from getting game-breakingly OP, which is fine. I can picture myself (and most players, honestly) treating this like a kid in a candy store with all of the early possibilities, and then running into issues like "how are you going to build Pyramids, Oracle, Great Library and Colosseum while getting your settlers and districts out and trying to pump out classical-era riflemen?"
Dramatic Ages are going to be very fun with Hammurabi, racking up GA points from free techs while pushing the next era along that much faster (at least in the early game.)
I wouldn't be surprised if Babylon turns out to have a start bias away from coasts, and possibly towards desert (which would be debilitating enough to counterbalance a lot of the UA.) That wouldn't matter on an archipelago map, but ocean-crossing isn't as important on those either, so not a huge deal.
I feel like Owls will probably be best for Babylon, who will want to focus on trade anyway to make use of their early tech lead, will want good Suzerainities (which will also synergize nicely with the free first district buildings in addition to Divine Spark, which is my go-to pantheon regardless but is even more powerful in Hammurabi's hands) and will really get going in the later eras when Babylon's lead starts to peter out. Voidsingers would still be good for a different strategy, though, focusing on faith production to get that to snowball, supplementing science and gold (and culture) in the medieval era, and allowing you to more-easily faith-purchase Great Scientists in later eras.
I've complained (a lot) previously about the "turtle up and click 'til you win" design of Babylon in Civ 5 (which is my least-favorite design of any civ in the series' history) and I'm glad that they went the opposite way with this one - potentially super-powerful, but you'll want to optimize every turn and really think ahead.