My shower curtain is feels like fabric. I just put it in the washer when doing a load of towels in a cold wash on gentle.
Both fabric and plastic shower curtains go in the washer. Add vinegar if there’s lots of soap scum. Wash on warm.
OMG Washing machine
Warm water, delicate cycle in the washer, add about 1/2 cup vinegar to take off the soap sccum, I usually do mine with a textured fabrics like towels and washcloths. This is for both the plastic liner and cloth pieces of the shower curtain assembly.
Soak it in hot bleach water in your washer. Then put on spin and rinse and take out quickly and pop into dryer for 3 minutes. Take out and shake, shake outward. Hang on rings immediately or it will wrinkle. The shaking cools the plastic!