Min-Chae is a Kim and Yi-Jin is a Baek, which indicates that the two are not father and daughter. But is this meant to throw us off? Twitter users have long speculated that Yi-Jin could have changed his name. We've seen how the impact of his father's bankruptcy has followed Yi-Jin wherever he goes, and it's possible he changes his name to Kim because of this. But developments in episode 10 may suggest Yi-Jin changes his family name to Kim (and maybe even his first name as well) because it's his biological name.
In episode 10, Yi-Jin's father calls him after a hard day at work and consoles him by telling him he was "his greatest consolation 10 years ago and you still are today." Yi-Jin responds by saying, "Thank you for raising me to be who I am." Yi-Jin is 21 years old in episode 10, so his father's dialogue hints at the possibility that he was adopted at 13 years old (why else would he have been his consolation 10 years ago instead of 21 years ago?). It may also explain Yi-Jin's general hard-working and humble personality in the face of hardship despite the fact that he was seemingly raised with a silver spoon in his mouth.
If Yi-Jin is, in fact, adopted and he decides at some point to change his name to his birth name, this could explain why Min-Chae never recognizes his name as Baek Yi-Jin. It also adds another layer to the moment when Jae-Kyung asks Min-Chae if she "knows" Yi-Jin in episode four. She may have been asking if Min-Chae "knows" her dad's name was once Yi-Jin.
