Lions and eagles are Old World symbols of power all right, but that doesn't mean squat as such. For all we know, the lion throne would have been war booty from the adversaries of the eagle people, or the entire court decorated with symbols taken from defeated opponents...
The soldiers could have come from just about any nation - or been generic mercenaries used by all sides. Apart from the Freemason symbols on their foreheads, they wear purely "functional" things (armor and weaponry), and those would probably be for sale to all sorts of bidders; essentially, the AK-47 of the Post-Atomic Horror.
As for governments, those could have blinked out for a few decades and then recovered, their verified later existence being no proof for their existence in 2079.
We can speculate along two lines, basically. First, we can assume that this court was typical of a prominent faction in the PAH era, with easily recognizable common characteristics that Data is giving his nod of approval to. Second, we can assume that such courts were a dime in a dozen in the PAH era, not specific to any particular power, and the ragtag nature is what Data finds accurate. But the second option probably couldn't justify the comment "very, very accurate"! Data would rather go for "quite true to form" or somesuch...
Furthermore, if we assume the court is typical of one power, we can also assume that PAH would not have been a completely global phenomenon. There would be places like this somewhere on the globe, and places mounting peaceful interstellar expeditions elsewhere.
We can see the court is multicultural, which is bound to confuse the issue. OTOH, the court is a hierarchy, with different ethnicities playing different roles, which could be taken to hold significance. The judge is Caucasian, as are his armed henchmen - does that mean a "western" seat of power? OTOH, the judge is Q, and his "true" face could be something different altogether - perhaps it's not the Asiatic gong-players who are subservient slaves from defeated nations, but the palefaced audience instead? Perhaps the soldiers are Danes in a foreign court, members of a subhuman breed who get to hold guns because the court has nothing to fear from these docile drug addicts?
As for the ECON, this organization could be just about anything in ST:FC, the only place where it gets a mention. Perhaps the people of Montana are afraid of East Coast Orthodox Nation bombers? Or that the Ecological Nihilists are going to go logger-hunting again? The fact that Picard denounces any connection to the "Eastern Coalition" can be connected to ECON, or then not; Picard may deduce that the vast Eastern Coalition somewhere across the ocean was the likeliest enemy of Montanans in WWIII, but Lily might have faced a more fractional reality where her actual opponents came from a wholly different, more local organization just across the state border.
Bottom line: we know nothing, and every interpretation is a valid one. But it does seem that the classic West survived WWIII basically unhurt, all its landmarks still standing, its values upheld, its citizens conquering space in this very era. Perhaps interpretations that keep the PAH out of North America at least, and perhaps the whole of the traditional West and more, should be favored?