If I can see a specific PC, Supervisor or Cost Object in the FRC tool, does that mean I am responsible for doing the monthly review?

Check with your supervisor to see (1) if they have assigned the FRC Reviewer role to you (indicating that you are only seeing the cost objects that you are responsible for) or (2) if they have not assigned the FRC Reviewer role to you, which PCs/supervisors/cost objects you are responsible for reviewing.