Well in Korea and numerous countries, gay marriage isn’t allowed or sometimes even legally penalized. So in those countries, some couple have to use adoptions to finally be part of the same family.
Else, if one of them finishes in the hospital for exemple, their lover isn’t allowed in nor to participate in the decisions. The hospital might even end up calling the person’s birth family, who might be homophobic and not want the sick person’s best interests.
Or if one of the lovers die, the one living cannot inherit. Those are just a few exemples.
In the case of the story, if the MC ended up dying and the ML hadn’t adopted them, Sol would get taken away, ML wouldn’t be allowed to keep her cause he wouldn’t be recognized as her family.
So yeah, alas, adoption is sometime the safest solution. Legally, it could be considered incest. But let’s be real here: in this story, they met as working adults (so no growing up together as siblings or cousins or something). There’s no grooming (so no abuse or possibility of abuse due to it), there’s no genetic link and they’re both men (so no reproductive risk for a possible offspring).
So it is incest only in legal name, and only due to a law that still discriminates people’s gender and sexuality

But isn’t that incest??? Idkkk but they’re so cute