Looks like Haebom adopt Ho as his son. Saw the word father there. Cause i don't think you're adopting a brother? So Sol will bear the Kwon surname. That's the reason why author didn't name the baby until the end. So that they all could be a family. Sol adopts Kwon surname from Ho. Ho adopts Kwon surname from being adoptive son of Haebom (legally)
In this case I think Sung Ho turns legally into his son, I have seen something like this before, in some countries where same sex marriage isn’t legal they one of them adopts the other so they can be family, being family legally allows for them to take care of each other if anything happens, be it medical procedures, or even if one of them dies this allows the heritage( I dunno how to say this, English isn’t my first language but I mean the money and possessions) to go to the other person. I may be wrong in some technicalities but I hope this helps.

I love the story so much huhuhu! I’m gonna miss waiting every week for an update grrr! Btw can someone enlighten me? I know same male-male marriages aren’t legal in SK but are they brothers in paper now? I’m just trying to figure it out. Sorry for the question!