President Moon Jae-in said it was not a gender issue.
Dozens of protesters dressed up in flamboyant jumpsuits and masks worn by actors in the Netflix show.
She reportedly walked out of the store while still wearing the clothes, before assaulting staff members who came after her.
They said it was an honest mistake.
Nineteen foregin aircraft entered its airspace on Tuesday.
Hongkon Kim is wanted on three separate charges dating back to 2017.
"It's very unprecedented that the Ministry of Foreign Affairs comes out apologising like this."
Anna Kim, her toddler and her mother were all killed in the natural disaster.
Anna Song was the Director and Treasurer of the New Zealand Chamber of Commerce in Korea.