Multiple private companies offer, for a fee, to "register" names or titles of ownership for stars. These companies are not affiliated with any official bod,y and their services are not recognized by any academic organization. The International Astronomical Union, which maintains star catalogs, disassociates itself entirely from the concept.
While some unusual stars do have names, often the names of their discoverers, the vast majority of them are known only by catalog designations that cannot be purchased or sold. Businesses that offer to sell or name stars, or which purport to sell real estate in space, have no authority to do so.