None of that other stuff is relevant to domain name. The "Amazon" camping shop in Pikesville, Ky doesn't need to change its business name (though they probably should if they set up shop in Washington state).
> What's the family business then meant to do, cease operating their family business under their generations-old family name?
No, just take another domain name. Millions of people have probably landed on nissan.com while trying to research a car, and been confused and frustrated in disbelief. And this is because one guy with the name "Nissan" beat them to the punch in the early domain-name grab. He could have just as easily taken uzinissan.com and enjoyed his 11 hits a month on that domain.