Yes this is possible. I have done it with both repurchased plates and reassigning existing plates.
Assuming you don't have the original black plates, then you would need to acquire a pair of matching plates and follow this procedure:
https://www.dmv.ca.gov/vehindustry/vin_memos/vin2009/09vin09.pdf
If you have the black plates stashed away and can prove that they were at one time assigned to your car then you foloow this (free)procedure.
https://dmv.ca.gov/vehindustry/vin_memos/vin2009/09vin20.pdf
In both cases my local DMV office was clueless and it took numerous phone calls to the processing center in Sacramento for them to finish the assigment or reassignments. Carefully check over the forms. Insist on a full set of copies for your records. If you bring in plates, they will check to see that they are authentic. The procedure states that they have to verify that they are metal and not aluminum. Sadly I had to bring in a magnet, then it was another 10 minutes of convincing them that magnets don't stick to aluminum...
It is a test, but can be done!
Dean
60 Healey with original black plates (yeah)
59 Peerless with yellow/black plates (YOM program VIN20)
71 Lotus with reassigned blue plates (VIN09)