From Gamesverse Wiki
There is both a Parliament directly elected via proportional representation from the populations, and the Council of Representatives, where people appointed by the dominant polity of Administered Worlds sit to represent their interests, as well as individuals from the various wings of the armed forces and the TSAB bureaucracy. The Chair of the Council passes between member states in a cycle; the Speaker of the Parliament is elected by the Parliament.
The Council devises policy; the Parliament ratifies, examines, and proposes changes to it, and can refuse to pass any bill. Local worlds have their own governments, although to be an Administered World you must have a democratic system in place which fits certain criteria for being fair, honest, and open. In practice, more power on a day to day basis rests with the Council, which is one of the demands of the smaller worlds, who are incredibly out-voted in the Parliament.
At least on some of the more integrated worlds, they'd synomised the members elected to the TSAB Parliament with their own government; others retain an independent democractic body.
Speaking generally, the TSAB is a more-integrated EU for the Administered Worlds. There is a common military, and there exists an optional common currency (which not all worlds, especially the ones who carry out less trade, are part of). Planets retain self-governance within the over-riding laws of the TSAB, and are responsible for their own legal code (which, again, has certain standards; member planets can be fined if their legal system does not hit necessary levels of judicial independence). The TSAB can intervene in extreme circumstances, but it tries to avoid having to commit forces to Administered Worlds, and would rather use soft, economic power. And things like threatening to cut all grants to a world which is persecuting its Belkan minority and, if they keep on doing it, threatening to issue a warrant for Crimes Against Humanity in the Dimensional Criminal Court (and then actually doing it, to the extent of Enforcers grabbing the politician in a surgical strike) works wonders.
So, yeah, to describe it as "a pro-integrationist vision of the EU" would be pretty accurate.
There also exists a set of Aligned Worlds, which exist in much the same condition as Norway does to the modern EU. They're formally independent polities, but they have some degree of legal standardisation (which varies), and many Aligned Worlds are currently in the stages of negotiating and hitting the entry criteria for membership.
For example, the world in which Southern Orussia, from Sound Stage X, exists is, in Game Theory, an Aligned World rather than an Administered World; it's too bogged down in the rather nasty civil war for membership, and looking at what we know about that war, even the TSAB would object to the use of child soldiers that young.
The TSAB will intervene if an Administered World fails to live up to their human rights standards, starting with soft pressure and rising up to "twenty Enforcers teleport in and arrest the entire ruling party while they're in session". The TSAB has, for the Administered Worlds, passed that legal threshold where power descends from the higher polity, rather than being granted in abeyance by the lower one. You will get a notable percentage of the population who will go "Yeah, I'm from Mid, but I'm just a TSAB citizen, you know" (and others who won't, still).
On Age of Service
The TSAB in its current form has been around for a grand total of seven or eight decades, and is trying to manage several hundred worlds. And by that I mean patrol the borders to stop the petty little fief-kingdoms around the edges from raiding them, repair the damage done during the Warring States Era, find and seal all the Lost Logia lying around, regulate the governments to keep standards of living at acceptable levels, capture dimensional criminals who strike out on their own, keep stability in the areas they control and stop it dissolving into another mess of fiefdoms and kingdoms and states that are mostly at war with one another, provide support to the Unadministered Worlds that are at Third World level and are trying to rebuild up to the point that they can join, stay friendly with the Allied Worlds that refuse to actually join them completely...
...and they're doing this across several hundred worlds, with perhaps a quarter of the manpower and half the ships they really need to do so; there are never enough mages, especially powerful ones; new planets are still joining so they're actually expanding (though at least they're getting more manpower faster than they're getting new territory); every so often they find another set of Lost Logia that aren't just really powerful engines or whatever, but which can actually destroy planets; the alternative to the TSAB keeping the region stable is the anarchy and chaos of the Warring States...
...and basically, if a couple of AA-rank nine-year olds say "hey, we want to help!"? They can't afford to say "no, wait eight more years and then you can come and do stuff for us". Because as long as those nine-year olds pass a psych evaluation - which they set high, because nine-year old brains are simply not developed enough to make those kinds of decisions, so only the most mature are grudgingly allowed through - then even with the mandate that they have to be kept to cases that have a very low probability of life-threatening danger and so on, that's two AA-rank mages worth of work covered. Which is the kind of thing that would take an entire squad of lower-ranked mages. The vast majority of mages in Dimensional Space do not have the potential to break A-rank. And while most people can theoretically get there, the number that get even that far is still small. TSAB forces average around high C-rank, with most new recruits being D-C and the more experienced members being C-B. A-rank mages are the cream of the crop, and while they're not ultra-rare, they're not exactly common, either.
When faced with a problem that you don't really have the manpower, technology or budget for, you can't afford to keep capable recruits off duty because they're young - a fact made even more knotty by the unpleasant cultural remnants of caste systems in some parts of Dimensional Space, encouraging people to go into the forces young and basically saying that they were born to be soldiers, with no expectation of ever being anything else. Which is another thing they're trying to get rid of, but it's slow going. The TSAB can and does try to make sure that they keep danger levels for children in the Bureau to a minimum, but as long as they're doing relatively safe work, that's two AA-rank mages worth of safe work that a squadron of older, weaker mages doesn't have to do. Which means they can go and patrol the borders, or hunt down a gang of dimensional criminals, or arrest a bunch of politicians on an Administered World who've been letting required standards of living slide.
If they could avoid using children, they would. In a heartbeat. But they don't have that option, and so they make sure that the kids are sent to safe, non-dangerous cases wherever possible (though of course sometimes they're the only mages in the region), and try to see to it that they get experience and tutelage doing useful, vital work that won't get them killed - helping set up an infrastructure on an Unadministered World, perhaps, or tagging along to bust some smugglers that intel deems unlikely to have much combat power. It's a crappy, crappy choice and they're already trying to phase out the younger members of the Bureau, only allowing in the ones so talented that they can't afford not to have them. Give it another couple of decades, and hopefully they expect to be able to put a solid age bar of fourteen to fifteen in place, which they can raise incrementally from then on until it hits sixteen or so.
But for now, it's just a question of manpower to territory.
On the Military
One thing has to be remembered here; that the TSAB is itself the post-apocalyptic remnant of other societies. This informs basic things about its structure. The TSAB began as the alliance of several post-Belkan remnant military blocks, assimilated the Saint Church bureaucracy along the way, and the civilian stuff is flesh over bone and muscle. Why is Disaster Rescue a military wing? Because that's what the military did in the post-Olivie chaos, because there was no one else to do it. Why is Environmental Preservation a military wing? Because when terraformed biospheres are collapsing due to a collapse of civil authority, you send in your engineering corps to try to stop the entire place dying off. Why is the FBI-equivalent a military wing? Because... well, it's at least partially because as Brits, Aleph and I am basing them quite a bit off the French gendarmes, but in-setting, it's because your dimensional criminals have historically been warlords who can have entire country-sized bases of operations on planets, and your police need people with the equivalent of tanks and gunships to take them down.
In quite a few ways, the TSAB is what you'd have got if at the end of Fallout, the NCR and Brotherhood of Steel had fused. There have historically been other schools of thought and developments, but the TSAB as it stands is akin to a hardened society. And it's further complicated by the fact that the modern TSAB is also a trans-planetary organisation. Disaster Rescue aren't the firefighters; they're provided by the native world. Disaster Rescue are the ones who help when there's a major natural disaster, when an oil spill threatens entire coastlines, or when a reactor threatens meltdown.
(Which, yes, does cast canon!Subaru's desire to join it in a slightly different light. It isn't purely a philanthropic thing; Disaster Rescue is also pretty glamorous in its own way.)
Hence, it's easy to under-estimate the TSAB size. If it really came down to it (like if they found another, aggressive, post-Alhazredian empire in Dimensional Space of their size), they could call up the Self Defence Forces and other assets of member worlds, and increase the assets available to them by at least an order of magnitude.