I’ve looked into it, and it appears that adult adoption does exist in Korea. The adopted person is legally added to the other person’s Family Register. Although they legally enter into a "parent-child" relationship in the eyes of the state, same-sex couples use this method to secure rights that would otherwise be denied to them: sharing a surname (the adopted partner can legally change their surname to that of their partner); inheritance rights (in South Korea, if there is no will—or if the biological family contests it—a same-sex partner inherits nothing; by becoming a "legal child," one is automatically granted the right to inherit assets); and medical decision-making (it allows partners to sign hospital consent forms during medical emergencies—a right otherwise strictly reserved for legal family members). Sorry if anything was wrong, I'm using a translation tool o.o but I assume this was Mr. Kwon’s strategy, making use of the tools available for them. I didn't know that
Yes, Mr Kwon adopted him so they are legally bound to each other in a conservative country like South Korea. (I'm bad with names) Imagine something happened to Sol's father later down the line. She would be left without any family around. Where would she go, an orphanage? Or sent back to a mother that didn't want her? Now, because of the men having that "family relation", Sol could just stay with Mr Kwon as he would be considered next of kin without any legal battles or anything.
They probably went with Kwon's family name since he's the older one (very age focused culture) or simply because it's not like Sol's father is too attached to his name, while Kwon's used in many places due to Mayor status and stuff like that.
It's not like they will go around and tell everyone they are father and son. They will act like a couple, live like a couple and be a couple. It's just some legal documents in lieu of gay marriage not being a thing to showhow serious they are about one another and to be able to act in case if emergency.
They re just dribbling the system that's rigged against them.

My heart goes to every queer couple out there that can't legally get married in their country :((
Just imagine the panic when one of your spouse got into accident and you can't sign the paper on their behalf cause you're technically not their legal guardian... Even worse if their family disowned them, you truly only have eachother as your own family member YET they're not legally recognized by the state so they can't make legal decisions for their partners...