At least the United States, unlike Denmark and much of the rest of Northern and Western Europe, you are not required to get governmental approval for names for your kids. My own children have names, a hypenated surname and Korean middle names, that almost certainly wouldn't have passed muster without elaborate efforts under Danish style laws. Government can go too far, and we manage just fine without limiting ourself to elaborate name regulations.
For what it is worth, this isn't an exclusively European problem. Korean regulation of names and regulation on marriage based not on bloodties, but on having similar names, is outrageous, and both the Koreans and Chinese generally have way too few surnames in common use.