So after the domain has been bought, the company can obviously see who bought it and find out your true identity. Is this a problem? (an inevitable one).
You'd be surprised how prevalent fake identities are in this business.
Years ago, I owned a lot of domains. Some very good ones, in fact.
I had someone approach me to buy the domain, claiming they were a tiny 2-person startup.
Some Googling and talking to friends who owned domain names revealed that they were actually representing Microsoft.
It's very common for even multi-billion dollar corporations to pretend to be broke students/entrepreneurs/non-profits to get cheap prices on domain names.
