(I’m not sure if this is an actual spoiler and if I’m remembering correctly) Cecilia is posing as a lesser known baron’s son and that’s also why she has a different last name and her family is the one that doesn’t know about whatever she’s doing in school. It was all arranged by Gil so some of the top brass of the school know her identity and are keeping it hush for them. Plus its also probably so that she doesn’t have to repeat school as Cecilia, lol
Oooo, now an actual spoiler
It becomes clear in the possible next chapters because after this the prince brings Cecil to a private room and interrogates her about the handkerchief. She answers that she’s Cecilia’s friend but is shot down by the prince who says that it didn’t make sense. Then she remembers that she was currently the son of a baron while Cecilia was the daughter of a duke and that not only were their status gap huge but they were also opposite genders so it was even more unrealistic for them to associate.
Gil comes to the rescue and says that he was his friend and therefore also friends with Cecilia after visiting their house. And the story continues