Back in ancient Korea, only the elite (land owners) in Korea had last names. It seems a little weird to imagine but back then it was totally normal!
Then Korea hit the 19th century and started to get rid of their class system. People socially below the elite were allowed to adopt last names - and here's where we see the birth of the "Kim, Park, Lee" families.
So if you're allowed to pick any name you want in the whole world, of COURSE you're going to want to pick a royal name! Who knows what that can do for you socially in the future :)
Not all Lees are part of the same family though! Each 'clan' has a distinct family history, even though their last names are the same!