The International Astronomical Union
acts as the internationally recognized authority for assigning designations to celestial bodies (stars, planets, asteroids, etc.) and any surface features on them.
and they say:
The IAU frequently receives requests from individuals who want to buy stars or name stars after other persons. Some commercial enterprises purport to offer such services for a fee. However, such "names" have no formal or official validity whatever.
As an international scientific organization, the IAU dissociates itself entirely from the commercial practice of "selling" fictitious star names or "real estate" on other planets or moons in the Solar System.
Accordingly, the IAU maintains no list of the (several competing) enterprises in this business in individual countries of the world.
From Ask an Astronomer at Cornell University:
Unfortunately it's not really possible to officially "purchase" a star for someone, although many people think so and there are companies that advertise this service.
The International Astronomical Union (IAU) is the organization that is responsible for naming celestial objects, and they don't take requests from anyone (even astronomers) for star names.
There are places where you can "buy a star", and you will get a certificate and star maps, but these names are not official and are not used in astronomy. In fact, the companies that do this don't even make an effort to communicate with real astronomers on the subject, they just keep a big book or spreadsheet with everyone's name in it.
Most people in astronomy, especially the IAU, consider the "star purchasing" idea to be a huge scam that is taking advantage of people's love of the sky to make a lot of money.