There are specific Korean dynamics in play that I wish another person had gone into but one thing to know is that there is a clear pattern with family names (in Korean, family names come first so they're not properly last names): in places where family names have been around for a while, there are much fewer of them than in places where family names were newly introduced or are in the midst of being introduced. Unless new names are being added (which is rare), there is a mathematical tendency for the number of names in a culture to decrease and, through founder effects and stochasticity, some names invariably become quite popular.

I go more into this in my older answer:

• As wives traditionally take their husbands' surnames, does that mean there are fewer surnames than in the past?

And here's a good thread with links to a lot of my other favorite last name threads (and also some of my favorite tidbits I think I only wrote up in that thread itself):

• When did surnames become a "thing" (i.e. a hereditary legal requirement)?

These post mainly focuses on the Latin West, but as far as I know, the system was in broad strokes similar, and generally centuries earlier, in East Asia.