IT all depends on the contract. If your customer refers to special requirements in its purchase orders, such as a supplier quality manual, and those requirements include PFMEAs, then you're obliged to comply. On the other hand, if there is no such documented requirement, your company would be within its rights to revisit pricing in light of new requirements. Whether or not this would be advisable depends on individual circumstances.
As far as who "owns" a
PFMEA is concerned, the organization that executes and documents the process "owns" it, again unless there are contractual stipulations to the contrary. No different from any other processes. I'm not sure what difference it makes, though. Why does this concern you?