South Korea has that problem right now. 20+% of people have the surname Kim [0]. 14% have the surname Lee [1].
> By some estimates 30 to 39 percent of Vietnamese people bear this surname.
A: Did you hear the news about Professor Kim?
B: Professor Kim, our department head?
A: No, Professor Kim over in Computer Science.
B: Oh, that Professor Kim, whom we met at the New Year's party.
A: No, that's a different Professor Kim. This one you never met.
B: Anyway, what about this Professor Kim I never met?
A: He got engaged to Miss Kim.
B: Our department secretary?
A: No, Miss Kim in the president's office.
...and so on.
The real problem by the way starts with multiple persons with the same or almost identically sounding names. My own name (Kang Seonghoon) for example is shared with at least 100s of other Korean people, including one famous singer who got arrested later ;-)
Speaking of having identical names, the one that cracks me up is 김정은.