There are a few different things here, so let's unpack it a little.Maybe this was answered earlier in the thread, but that's why I'm a lazy sloth and don't read.
How do breweries make these sweeping statements about aging their beers? For example, Goose Island's famous, "Will develop in the bottle for up to 5 years, or Cantillon's 10+years whatever-is-on-their-labels". How do these breweries come to that conclusion that the beer inside will develop in the bottle for that long? Is there an accelerated aging process they use to determine how long a beer can be cellared before it starts to go downhill?
0) Is making claims of aging potential useful?
No. Everyone's palate is different to the point where these things just aren't meaningful. They're basically just marketing. Though getting some insight into what the brewers think about aging their beer is sometimes useful on the margin, it's so woefully inadequate compared to personal experience as to not even merit mentioning alongside experience.
1) Are American breweries' claims of aging potential based on anything?
By and large no, they're just marketing bullshit. Deschutes is famous for this, their "best after" thing is insulting to anyone who knows what's going on. Goose Island at least can claim that they have that much history aging BCBS, but my bet is that they started putting that on their bottles long before they actually had any experience with it. Though 5 years seems right about where BCBS starts falling off, so at least they were about right. RR's markings are pretty aggressive too, but, again, though they have the experience with it, it was basically just guessing when they started doing it (and afaik they haven't changed their dates). And, as I said above, this will fluctuate wildly with personal preference, and to a lesser extent, storage conditions.
2) Are Belgian breweries' claims of aging potential based on anything?
Yes! Cantillon has been brewing more or less in the same way for over a hundred years. Even their relatively new beers have been around for 10-20, and those are such minor variations on old styles that how they age can be extrapolated pretty well. This goes for pretty much all of the Belgian breweries, because even the new ones are making those classic styles in the same way (and often with the same lambic). But, as before, how much age you like on these will vary considerably.
2a) Aren't you being a little bit dismissive of the Americans' claims? It's not like "stout" is a new style.
Yeah, but "barrel-aged stout" sure is, it's been around for 20 years at most. And American Wild Ales are even younger, and the experience brewing and blending them just hasn't built up to the same levels as the Belgians yet. Also, a lot of these beers have adjuncts in them that are pretty new, or processes/bugs that are new. Knowing what will happen to them with age will take some time. I think that, by and large, BCBS or Vinnie's stuff is there now, but things like Black Butte XXV just aren't. Even for a brewery that largely knows what they're doing and has been for a while, but does it with a new product (like GI with King Henry or Bramble) making those aging claims is little more than guessing. Which is why I said that they're "mostly bullshit".
3) Is there an accelerated aging process?
I want to be emphatic here. You will sometimes see claims that aging at higher temperatures makes things age faster, but this is misleading at best. In general chemical reactions proceed at higher rates at higher temperatures, but those increases:
1) Aren't linear (that is, an increase from 50 to 60 is, in general, not going to speed up a given reaction by the same amount as an increase from 55 to 65).
2) Aren't the same (that is, not every reaction increases at the same rate with temperature).
Also, this is completely ignoring that some reactions have minimum energies needed to happen, so a reaction could basically never happen at 55 but might go like gangbusters at 80, while the other ones have barely increased.
As a demonstration, production of the "cardboard" flavors we all know and love from over-oxidized beers are strongly dependent on temperature, such that they take years to form at 55 but hours at 110.
So there's no way to speed up the aging process such that it'll be a meaningful replication of the natural one.