A Foreign Experts Certificate, (FEC) is required before one can be issued a Residence Permit.
The FEC process has moved online, over a year ago in our city.
Typically, the employer submits all the requisite data online to the local SAFEA (or whomever processes it in your city) Office and they then make a determination of qualifications.
This usually only takes one day if they deem the employee qualified to be a "Foreign Expert". If all is approved, the employer then submits all the hard copy paperwork for processing the FEC, (which takes about a week).
The online submission process is quite straightforward. One simply types in data and uploads files (photos, for example). The database system has the usual error trapping one might expect; mandatory fields, etc. I am unaware of any glitches that would be caused by entering different employers.
Many schools do not issue FECs to the employee. The employees never see them. A while back the FEC needed to be carried for identification and to get discounts on lodging, travel, etc. That has long gone away.
The FEC is supposed to be returned to the employer when the employee leaves. Many never did that. Thus, some employers, perhaps out an abundance of caution, or an excess of need for control, no longer provided them to employees.
I can't explain your school's position. They should have you current FEC.
Now, some employers have skirted the FEC process by hiring "alien workers" under the labor bureau, but those are rare cases.
If your school has access to the database, (SAFEA licensed schools have a password protected account) then it should have no problem entering the data unless there is some employee operator error.
I hope this helps.