I am sure she meant like as in a friend based off her actions, but I wonder how the grammar worked in Korean. Do they have a word that means like in both a platonic and romantic sense like in English, or was this explicitly a romantic like that when translated causes a bit of confusion.
Always happy to chat about Korean!! The verb used in Korean is 좋아하다 which can mean "to like" both romantically and platonically, just like in English. If it were explicitly romantic, you can be sure I would've mentioned that.
You wondered about the grammar; the grammar doesn't influence the fact that "to like" is used, but because you asked, the sentence used here is "저 백우진 선배님 좋아하니까", and the sentence is structured as a follow-up to the previous dialogue ("Don't tell me how to feel" etc).
저 = I
배우진 선배님 = Baek Woojin-seonbaenim
좋아하니까 = the verb 좋아하다 conjugated with a suffix that means "because", or "I said".
So if I translated the whole dialogue a little more literally and less naturally: "Don't you tell me how I feel, because I like Woojin-seonbae." 😊