I don't think it is translated as much as it is that every Korean family actually has their own Hanja.
According to Wikipedia's List of South Korean surnames by prevalence
can be one of four Chinese characters:
柳, 劉, 兪, 庾
Most Koreans have Hanja names & they would certainly know which their surname is.
If you check out Wikipedia's Appendix:Korean surnames you can see that
is the Southern Korean equivalent of
Do you know what it is? It is a Korean newspaper printed in 1920. It is printed in Korean language, not Chinese. The Chinese characters in that newspaper are actually “Korean Chinese characters”, known as hanja in English. Hanja is still the official script of Korean language, alongside Hangul.
Hanja has not been abolished in South Korea, but few Korean people use it today. Most of Korean people only write Hangul today, except their names. Korean names are still in Hanja.
Have a look at South Korean identity cards. Korean names are still printed in Hanja. As a result, we never translate Korean names into Chinese. We only copy the Chinese characters into Chinese.
Therefore, the real question should be:
Why is the Chinese character 刘 (劉) pronounced 유 (yoo) in Korean?