Static shielding bags for PCB protection
However, does this stay true if for example a PCB is wrapped in plastic or paper before being inserted into a metallic static shielding bag which is then sealed?

No

Would this material be able to damage the PCB even though it is contained inside the faraday cage?

Yes
They primarily prevent triboelectric charge buildup on the body of a person handling the contents from being transferred by conduction. They also prevent such transfer from other materials that the might come into contact with such as mechanically-protective packaging materials used for shipping and storage.