How do you know that the Federation President's office is in Paris?
In medieval times the capital of the eastern Roman or "Byzantine" Empire was clearly in Constantinople, with the main imperial offices in the Great Palace or in the Blachernae Palace at various times. But in western Europe in medieval and early modern times monarchs often traveled around their realms with their courts and administrations, stopping at various manors and castles they owned, or at various cities, or in the estates of powerful nobles, or in monasteries.
By comparing the dates and places of various charters issued by the chanceries of a specific monarch, historians can trace the monarch's movements from place to place within their realm.
In ancient times the mighty Persian Empire of the Achaemenid Dynasty had four capitals and the court moved from one to another: Babylon in Babylonia, Susa in Elam, Pasargadae and later Persepolis in Persia, and Ecbatana in Media. In the Arsacid or Parthian Dynasty the capitals included Susa, Nisa in Turkmenistan, Shahr-e Qumis, Asaak, and Ctesiphon in Babyonia as the winter capital. In the Sassanid Dynasty the capitals included Istakhr in Persia, Gor or Firuzaband or Ardashir-Khwarrah in Persia, and Ctesiphon in Babylonia as the winter capital.
Here is a link to a list of countries with more than one capital at the present, having different capitals for different purposes or different capitals for different seasons.
And possibly Federation institutions like Starfleet move around from place to place on Earth from time to time so that different places can have the honor and income from hosting them.
Whenever Stafleet Academy is seen, it is seen in the San Francisco area. Whenever Starfleet Headquarters is seen, it is seen in the San Francisco area. Thus Starfleet Academy and Starfleet Headquarters might be permanently located in San Francisco. But it is possible that they periodically move from one location to another, and by coincidence they have only been seen when they were located in the San Francisco area.
The federation President's office is seen to be located in Paris, France in Star Trek VI: The Undiscovered Country and decades later in DS9 episodes "Homefront" and "Paradise Lost".
The Federation President's office was located on Earth in the city of Paris, France. (Star Trek VI: The Undiscovered Country; DS9: "Homefront", "Paradise Lost")
Possibly the Federation President's office periodically moves from one city to another one, and Star Trek VI happened to be while the office was in Paris, France, and "Homefront" and "Paradise Lost" happened to be when the President's office had moved back to Paris, France after being in other cities for decades.
With advanced future Star trek technology, it is even possible that the office building of the Federation President is moved from one city to another, like a spaceship that is usually docked, with only the identical foundations in each city being permanently located.
Similarly, the Federation Council is said to be in San Francisco in Star Trek IV:The Voyage Home.
In 2286, the Council convened in San Francisco.
So it is perfectly possible that the Federation Council travels from place to place on Earth.
So if the Federation President, and Council, and other government bodies move from place to place on Earth it would not be unprecedented on Earth, and possibly it would also have precedents on other federation planets.
What if the locations of the Federation Council in San Francisco and the Federation President in Paris - and possibly other Federation functions in other Earth Cities - is permanent?
The list of countries with multiple capitals shows that some countries have different government functions in different cities, so if the Federation Council is permanently located in San Francisco and the Federation President is permanently located in Paris that would not be unprecedented on Earth. And possibly there are also precedents on other Federation planets. And of course the Federation is Gazillions of times larger than Earth but has such vast speeds of transportation and communication that it can function, so having the Federation Council and the President's office permanently located a few thousand miles apart would not be a large inconvenience.