As I understand it, Korone has literally made herself sick before by refusing to sleep, so I get it when her chat worries about her. Still, you shouldn't flood chat with "go to sleep" messages.
Does she share birthday with her realself? That seems to be the case with the last panel. Interesting the friend isn't suspicious about that.
I think we'd have seen Teruko being annoyed about her crush paying more attention to her vtuber identity than the real her if that was the case. Sukoya was probably just happy about it being a big day for her oshi