And there are also businesses with identical names. But the basic idea was already established long before the internet. If you have a legitimate claim to a name, you have a legitimate claim to that name. There may be multiple entities with a legitimate claim to a particular name, in which case the first one that used it in a particular context gets to use it in that context. And if you think that someone is using a name you have claimed in a misleading way or acting in bad faith, you can sue them and let the courts decide.
The problem is that as you note, trademarks and company names are not unique, but domain names are required to be unique. So that n to 1 relationship between trademarks/names and domain names intrisically creates problem, how to allocate the domains when there are many equally legitimate pre-existing claimants. This is not solved problem the way you portray it, because domain names have this novel uniqueness requirement.
Of course this raises valid question if using names in this way at all is a good idea. For example telephone system and lots of banking stuff is based on simple numerical identifiers, and lots of countries have also some unique (numerical) identifiers for companies and persons. So there is fairly strong precedent for using assigned ids instead of names when uniqueness/specificity is required. But somehow we have jumped to the conclusion that for example IP addresses would be too confusing to average joe, and in attempt to hide them we have created even more confusing system.
Many countries already solved this problem with their ccTLDs decades ago. It only required taking the established practices and applying them to a new class of names. There are always some edge cases, but domain name assignment is pretty much a solved problem.
