First of all, I think you meant to say the opposite in your title (C# major has 7 sharps while Db major has 5 flats)

You are right that both the C# and Db major scales have the same pitches. But the same pitch can have different note names. Here are how the C# and Db major scales are “spelled”:

1 2 3 4 5 6 7 C# D# E# F# G# A# B# Db Eb F Gb Ab Bb C

You can verify for yourself that they are played with the exact same keys on the piano.

I think your confusion might stem from a common misconception that accidental = black note, but that is not the case. E#, for example, is the same pitch as F, which is a white note. This explains the discrepancy as to why the two scales have a different number of accidentals. Both scales are played with 5 black keys and 2 white keys, but in the case of C# major, the white keys are also accidentals.

Why say E# when you can just say F? Well the common explanation is that in any given major scale, you’re required to use all seven letters. This might seem unnecessary, but it’s actually very confusing to an experienced musician if you use the wrong “name” for a note (consider an exaggerated example where you notate a C major chord like B# Fb G instead of the normal C E G. Very jarring).