Can a counterfeit Shimano chain be used on an authentic Shimano drivetrain?
You can normally use chains from other legitimate vendors, be it SRAM or KMC or something else. You do not need to use Shimano™® chains only.
But no-one can tell you whether your chain is of sufficient quality. Only you can inspect it, measure it and try it. We cannot do that.
I am not even 100% convinced it is actually a fake chain, but I will just take is as a premise I am answering to.