The only place you leave your finger prints in China is at the scene of the crime assuming you are not wearing gloves.
Some banks require it.
Some hospitals require it when signing medical documents.
A school? No.
Some schools in China also try and use biometric readers for attendance. No.

Its likely some bullshit in-house thing with how they do business but you don't leave finger prints anywhere.

It would not be unheard of to have a print lifted and placed somewhere should the need arise.

You are in China.

Don't be stupid.

You don't need to have anything to hide in order to have everything to protect.