I got curious and looked it up but apparently it's not common in SK to adopt your partner because the court can technically reject you if it is obvious you are trying to find a loophole about marriage, which is why I'm surprised the author chose this route (but I also understand the significance of it in this story)
As per the document shown on the laptop, he did adopt ho as his son on paper to be called a family coz that's the only legal name coz you cannot adopt one as your mother or father or adopt them as your siblings. It's always as a parent and child. That's why Kwon said he was okay to be Sung Haebom (surname of Ho) but he was too old to be adopted as a son by Ho, so it was Kwon who adopted Ho on paper. (This is how I understand it)
They're main goal is to be really a legal family on paper, because there's no marriage available.

Hold up wait a minute did he adopt his boyfriend as his son I need clarification. I just really need clarification lol I wasn’t sure if he was adopting his baby or he wanted him to adopt him.