A capit
al is the city (or, in
Star Trek's case, the planet) in which a sovereign state's government is officially headquartered. A capit
ol is a building in which a government (or, more specifically, a legislature) is officially headquartered. A capitol is located within a capital.
Now, as far as the Federation sharing a capital planet with United Earth -- let's face it, this is pretty much inevitable. The Federation capital planet was realistically-speaking going to be
one of its Members' capital worlds. That it is Earth actually suggests a compromise between Vulcan, Andor, and Tellar, since Earth was the "honest broker" rather than a hegemonic power. In fact, United Earth had very recently been little more than a client state of the Confederacy of Vulcan.
Now, as far as the capital city goes -- this is flatly false, so far as I can tell. Unless I missed something very major in the ENT novels, the capital city of United Earth has not yet been established. We know from
ENT: The Romulan War: Beneath the Raptor's Wing that the United Earth Prime Minister maintained offices in Paris, France, in which he sometimes held Cabinet meetings. But we also know from the immediate prior novel,
ENT: Kobayashi Maru, that the United Earth Prime Minister maintained offices in San Francisco (which was the city which also housed the Coalition of Planets Security Council). I can recall no mention of what city housed the Parliament of United Earth. The actual capital city, if any, of United Earth was not revealed so far as I can recall.
As far as their respective capitols...
I'm honestly not sure what you're referring to.
The closest I can think is that you've gotten the Palais de la Concorde confused with the Place de la Concorde.
A Time to Kill and, later,
Articles of the Federation, established the Palais de la Concorde to be the Federation's capitol, housing both the Federation Council and the Federation President.
ENT: The Romulan War: Beneath the Raptor's Wing established that the United Earth Prime Minister's Paris office was located in the Place de la Concorde.
Now, the thing is, the
Place de la Concorde is a
real location.
Place, in this context, refers to a large public square in Paris, of which the Place de la Concorde is one of the most famous. Indeed,
Articles established that the Palais was built on the Place in part because of its historical significance.
Now, I tend to agree with you that it was a subjectively bad decision to reveal that the U.E. P.M. had an office at the Place, because it implies a certain Earth-centric mentality in the symbolism for the Federation capitol. However, we should bear in mind that we don't really know what the Federation capitol's exterior looks like -- it could full well be a hybrid of Human, Vulcan, Andorian, and Tellarite architectural styles.