I don't know about the legal process in Korea. But, in my country, you can add someone to your family register and you will become a legally bind family under an official breadwinner family member. In this case, it's probably like Haebom is the breadwinner, the holder of the register, and Ho and Sol are under his register. It's not really adoption per se but it's still adopting I guess. You can add like "brother" and "daughter" under the "how this person related to the register holder" column. I don't know why Ho has to change his surname to Kwon. But this is a general idea.
I don't know about the legal process in Korea. But, in my country, you can add someone to your family register and you will become a legally bind family under an official breadwinner family member. In this case, it's probably like Haebom is the breadwinner, the holder of the register, and Ho and Sol are under his register. It's not really adoption per se but it's still adopting I guess. You can add like "brother" and "daughter" under the "how this person related to the register holder" column. I don't know why Ho has to change his surname to Kwon. But this is a general idea.