Q: Who is legally responsible for naming objects in the sky?
A: The IAU is the internationally recognized authority for naming celestial bodies and surface features on them. And names are not sold, but assigned according to internationally accepted rules.
Q: What does this mean in practice?
A: Simply this: Names assigned by the IAU are recognized and used by scientists, space agencies, and authorities worldwide. When observing stars and planets or launching space missions to them, or reporting about them in the news, everybody needs to know exactly which location a particular name refers to. The names assigned by the IAU are those that are used. These rules are firm where claims of property could theoretically be made, i.e. primarily in the solar system (where also treaties negotiated through the United Nations apply). Terrestrial makers of international law have so far had more urgent concerns than creating rules for "buying" totally inaccessible corners of infinite space, so there is no written text that can be twisted and interpreted - just a plain and practical fact.
Q: But if I want to, can I buy the name of a star anyway?
A: Sure, there are people who will be more than happy to take your money...
Q: Can you tell me who and where?
A: Sorry, we are a scientific organization, not a branch of the entertainment industry. We cannot distribute addresses of enterprises selling fictitious goods.
Q: OK, I found a dealer myself; what will I get from them?
A: An expensive piece of paper and a temporary feeling of happiness, like if you take a cup of tea instead of the Doctor's recommended medicine. But at least you do not risk getting sick by paying for a star name, only losing money.
Q: But that name is unique, I understand?
A: It will be likely unique in that company's name list. Otherwise you can probably sue them. But there are more than enough stars for everybody who wants to buy the name of one. However, no countries, authorities, or scientists in the world will recognize "your" name for the star. Nothing prevents your or any other dealer from selling "your" star to anyone else. And just think of all the other stars in the Universe that also have planets with smart business people on them...
Q: My friends tell me the name is preserved forever?
A: Sorry, also not: The name you paid for can be ignored, forgotten, or sold again to anyone else by anyone at any time.
Q: But the company says their name list is registered with the National Library - isn't that a guarantee for authenticity?
A: Sorry again: Anyone can (in fact usually must ) send a copy of any published book to the National Library. Giving the book a number doesn't mean that the Library approves the contents or checks that no companies "sell" the same star to different people.
Q: Surely the courts will recognize the name I have paid for?
A: Try to contact your lawyers. Chances are that they will either laugh their heads off or politely suggest that you could invest their fees more productively...
Q: But what about the companies that sell pieces of territory on the Moon and other planets? Those are within reach, we know, so surely I own the piece that I have bought?
A: See the answer to the previous question. As a minimum, we suggest that you defer payment until you can take possession of your property...