During 5 legal employments over 10 years, I have only had an FEC in my possession for a few months, an event which got the FAO assistant who gave it to me in trouble.

I have always had photos of it. As I recall, all employers obtained it without my presence during the day that they obtained it.

If the school is being investigated for hiring practices, they probably need it in their possession to verify the legitimacy of their payroll. Individual teachers questioned by the police need the residence permit in their passport.

Even a Chinese passport is the possession of the state, so I expect that this document has the same status.