Several companies claim to let you buy or name a star for a fee, and we astronomers are often by customers of these companies if we will use their names or take a picture of their star. The answer is always "no," for various reasons.
Although companies "selling" stars or star names likely do keep track of who has named which star what, these names are unofficial. Research astronomers call objects after names approved by the International Astronomical Union. Outside of objects in our Solar System and stars with common names (like "Polaris" or "Betelgeuse"), stars and galaxies don't get anything approaching real names. They get names based on either the catalogs that list the stars and galaxies (such as Gleise 581, the star recently announced to have an Earth-like planet) or based on their coordinates (like PG 1115+080, a gravitational lens near the coordinates RA=11hr 15min, Dec = +8.0 deg).
When you send money to a company selling stars, you are buying a product from them. Often it includes a certificate, a sky map with your star marked, and maybe some information on stars. If you give the company a name, they keep it on record, sometimes promising to publish the names in a book. But you are not doing anything official, and often what you are getting may not be worth the money.
Let me give you an example. One year, a family member thought it would be neat to give me, an astronomer, a star. I thanked them profusely, and then named the star something like my "Totally Bogus Star." I refuse to say which company it was from, though.
The certificate I received came with coordinates of the star, as well as a star map. So, for fun, I went and looked the coordinates up in our catalogs. There was nothing at those coordinates! I tried several variations on the coordinates (perhaps the coordinates were old, or perhaps the star was moving fairly rapidly), but nothing came up. Finally, I tracked down the star circled on the star chart. It was pretty far away from the stated coordinates! And, as I knew it would, it already names from several catalogues produced in the late 19th and early 20th century.
I can guess what happened to the coordinates. The sky chart was printed out from some popular astronomy software; the software allows you to see coordinates of a spot by moving your mouse around on a star map on the screen. I suspect somebody just read those coordinates off. They were close, but not nearly as accurate as real star positions are known to be.
But the only thing that angered me was a booklet on stars sent with the certificate and star chart. It was full of horrible factual errors. It was clear that whoever wrote that booklet did little if any research into what they were writing, and certainly had no editor checking their facts. I find this upsetting, because (a) many of the errors are so outrageous that the tiniest bit of research would catch them, and (b) most of the people who read the schlock are not going to know that it is wrong, and in fact are probably more likely to believe it because it comes from an "authority," a company that deals in stars.
If you have "bought" a star or are thinking of doing it, I won't tell you not to do it. But realize that what you are getting is not official naming rights for a real star. In fact, much of what you are getting is likely very cheaply produced with little oversight or care about quality. For the price that these stars cost, you deserve better.