Yes it is true that Korean women do not take their husband's name and keep their own family name.
There can be three possible scnearios:
1. LOST might have been written by someone who didn't know much about Korean culture
2. it might have been that they did it for the sake of keeping things simple for them and the viewers
3. The couple adopted the American way and decided to take the husband's last name
I mean, if they were to have different last names, then some American viewers might wonder why, then they would need to think of a script and a portion of a scene where they would have to explain it, and perhaps they decided that would be more hassle than necessary...but like I said before, they writers may have just not known it.
Generally speaking, a lot of American shows have cultural misinformation like these, so don't be too surprised.
this is what is says on the LOSTwiki:
"When Sun was married, if she followed an old tradition of Korean women, her title would be Paik Sun-Hwa, wife of Kwon when the relation has to be known. It remains unclear why she followed the Western practice of taking her husband's surname. She may have just let herself be addressed as "Mrs. Kwon"; however, Colleen referred to her as "Sun-Hwa Kwon", so it can be inferred that she took Jin's name. "