Because Sylphy consider it Ariel's secret and thus doesn't want to reveal it without Ariel's approval.
Basically when Ariel want to go out and stop holding up the 'perfect student' appearance, she'll disguise herself as Fitz to do so, during which time Sylphy would go somewhere isolated like the library (hence why she found Rudeus there that day)
However, the disguise does not change the eye, meaning any observant person can tell the difference if they knew how Fitz normally look, thus the sunglass mean it's impossible to tell the difference.
As for why she didn't reveal herself as Sylphy, because she think Rudeus forgot about her (since he didn't recognize her even though she recognized him)
So her current plan seem to be getting close to him as Fitz first.
(Ariel already gave her approval both from the view of the princess, to gain a powerful mage like Rudeus as ally, and as a friend)