I think your mention of the EU and bankers in the question frames a good avenue to explore for your solution. The multinational organisational frameworks we have today are largely trade bodies - the EU being a perfectly good example, although in the case of the EU there is a good deal more integration than is strictly necessary between members of a trade treaty. The EEC which was the precursor to the EU did also originally grow out of a specific desire to foster co-operation among the European nations in such a deep rooted way that it made a repeat of WW2 impossible.
So I think that for the idea of such a world government to catch on in your world you may need 2 factors, though the importance you assign to each may not be equal: well connected trade around your world, and a legacy of horrific inter-state or inter-bloc wars.
Then for it to be practical you need reasonable communications among states. You say that your world is not Earth: in that case, you may be able to improve the ability of states to communicate by altering the distribution of continents to avoid having them separated by colossal oceans - at the time period you compare with on Earth there was reasonable communication within land masses - certainly within Europe and within East Asia (though communication between Europe and East Asia was poorer because of the harsh mountainous territory encountered in between, i,e, the Silk Road) - but almost no communication across the Atlantic or Pacific oceans.
As to how it exerts its influence, I suspect for it to work along the parameters you have suggested, it does not do so militarily to any significant degree. 145
Maybe a bit of UN-style peacekeeping, but mostly by legal enforcement of the members' treaty obligations backed up by sanctions where states go rogue. States would almost certainly still have armed organisations for defence of the territory (just in case...) and for controlling their own populace, but the main defence against aggression would be that the entire world would cut off the aggressor as a pariah state and inflict huge economic suffering. So, yes, in the end, bankers on steroids. But not just one clique of bankers like the <PERSON> on steroids, but a clique of all the individual rulers and heads of national banks who want to ensure their continued profits, and while some degree of conflict can be profitable large scale devastating wars may not be, and at the very least are hugely risky - you might lose...
This kind of thing works much better in a world where there are extensive trade relationships (globalisation, if you like) than one where every nation is more-or-less self sufficient maybe except for a few luxuries like spices from abroad, which probably also fits in with your concept of a world that has developed slowly to a state of maturity. 145