You file for California state unemployment online, you state that Media Services is your employer. (Look at the bottom of your pay stub, it has the proper info) and any other payroll like Cast and Crew etc. Media Services IS your employer of record.
It doesn't affect production companies, nor Media Services, it comes out of a state fund, to which media services and you paid into.
The production companies only fund their half of payrollll taxes along with you to MEDIA SERVICES.
I was asked on an unemployment interview about this "media services", and they responded, "oh you're in the entertainment industry, got it"
IMPORTANT DISABILITY NOTE:
If you want to file DISABILITY, look at your pay stub deductions, if it says CA SDI, then you file with the state, if it says CA VDPI. theres a number on your Media Services pay stub to call (its a privately funded disability insurance).
I get paid by Media Services, and have done both.