So it’s the capital of two states, but located in neither of them? That is truly awesome.
I wonder if there are any other examples of a city not being located in the territory of which it is, in supposed or actual fact, the capital.
Berlin is also simultaneously a state in it’s own right and the capital of Germany. During the Cold War West Berlin was an autonomous city-state under nominal Anglo-Franco-American military administration (ie not part of either Germany) while East Berlin was the capital* of East Germany.
*The Western powers never recognized it as the capital of East Germany, but did eventually open embassies there.
Don’t overlook the obvious. Washington, DC to Annapolis, MD is something like 32 miles.
Shannon County and Todd County, both in South Dakota don’t have county seats. Instead, Shannon County’s administrative center is in Hot Springs, SD, which is also the county seat of Falls River County, and Todd County’s administrative center is in Winner, SD, which is also the county seat of Tripp County.
However, DC and Annapolis are not adjacent in any meaningful sense. Getting from one to the other requires going though other communities not considered part of either city. Now, if Arlington, VA or Alexandria, VA were capitals, I would consider that adjacent, because the city limits (which are also the same as the territorial limits of the not-part-of-any-state district) press up against the limits of Arlington and Alexandria.
Providence and Concord are close enough to Boston to qualify as its outer suburbs. Many commute the distance to work.
Pretoria is the national capital of South Africa while Johannesburg is the capital of Gauteng province. The outer limits of the two cities touch each other, although the centre-to-centre distance is about 50km.
There are dozens of examples in the USA of county seats that are in adjacent towns. Blountstown and Bristol, Florida is the first that springs to mind, where the two county courthouses are less than a mile apart. New York City is the county seat of five counties.
I think that if we extend the search to US counties there will be far too many to list. (Though I’ll wager that Minneapolis and St. Paul are the among the most populous pairs.)
Oh, and I just noticed a rather glaring error in my OP. I actually do know how to spell Wiesbaden properly… Og knows I’ve been there often enough. 
Perth Amboy was one of two rotating capitals of New Jersey for a while after independence. It’s just across from Staten Island, which is now part of New York City, which was once the capital of New York. However, I don’t believe New York and Perth Amboy were ever capitals at the same time, and Staten Island wasn’t part of New York City until long after the capital was moved to Albany.
Wikipedia has an article on the matter.
Most of those examples are places that are the centre of their metro area but which have some independent status. So they are still within the area over which they preside. I think what we are looking for is more like another example from your link, Kingston upon Thames, England, which is (for most purposes) the county town of Surrey, despite having been wholly outside the county for nearly fifty years, since it became part of Greater London.
In days of old, after the US gained independence but before DC was planned and/or ready, the capital of the US changed several times. It looks like the definition of “capital” that is used in this case is the city in which Congress met - so if Congress packed up and moved, then they carried the capital with them as a matter of fact and law without needing to explicitly pass a law changing the capital. Some of these cities are in the Mid-Atlantic area and are close to each other as well as to some State capitals. Some of these cities are State capitals today - if they were state capitals too back then that would be interesting.
Former US Federal Capitals:
Pennsylvania:
Philadelphia (the former US Capitol Building in Philly is still standing and tourable)
Lancaster
York
New Jersey:
Princeton
Trenton
Maryland:
Annapolis
Baltimore
New York:
NYC
Not quite the same thing, but after the fall of the Ayutthaya kingdom in what is now central Thailand, the Siamese capital moved to Thonburi. A few years later, they mover it right across the river to what is now Bangkok. The Chao Phraya River divides the two, so you could say they are next to each other, except now Thonburi is a district of Bangkok itself.
It is my understanding that all of these cities fit the definition that you provide in the second sentence of your post: They are the administrative centre of an administrative unit to which they do not belong. It’s not just being part of a metropolitan area irrespective of administrative boundaries, they are also outside the administrative unit they serve, owing to their independent status.
This is, however, not at all unusual. It is very common in Germany, for instance, to have medium-sized to large cities which have county-level status (kreisfreie Stadt is the German term - a city which is “free” from being part of a county) even though there is an adjacent county that bears the same name as the city, and the county administration of which is located in the city and thus outside the city it serves. It’s a constellation that occurs relatively frequently here, and, I suppose, also elsewhere. But it’s rather to have that at a higher level (states, in U.S. terminology, rather than counties). The only examples I can think of now are Vienna until the 1980s, and Chandigarh, the latter having the additional oddity that it is the capital of not one but two states to which it does not belong.
A similar case would be Rio de Janeiro: There is a Brazilian state of that name, but until the foundation of Brazil’s new capital Brasília, and the relocation of the federal government there, the city of Rio de Janeiro was a federal district (analogous to D.C.) and not part of the state of Rio de Janeiro. The state government was nonetheless located in the city of Rio de Janeiro. The thing has changed, however, and the city of Rio de Janeiro is now part of the state of the same name (and its capital).
For a non-European example of two national capitals located relatively close togther, Kuala Kumpur and Singapore are only 375km apart .
Depending on how you view their status, Hong Kong and Macau are only about 65km apart.
And one of my favourite pieces of geographic trivia: London and Paris are only about 340km apart as the crow flies.
That’s true, but by European standards 340 kilometres is not that short a distance. There are other pairs of European capitals that are closer than that.
The Vatican City is the capital of the Vatican States, though: the States aren’t limited to the City.
In India the state of Andhra Pradesh may be about to be split in two, after which the existing state capital, Hyderabad, will be in the new state of Telangana. It’ll serve as a joint state capital for ten years after which, I assume, the rump state of Andhra Pradesh will have to choose a new capital.
Oh, I know by European standards it’s quite a distance, but people tend to think of London and Paris being a reaaaaaaallly long way away from each other - not a 90 minute train trip.
I suspect the English Channel and being on a separate continent has a lot to do with the perception the two cities are a lot farther apart than they really are.