I would guess they do, but the term plastic surgery is older than the material plastic. It is used here as an adjective meaning ‘capable of being molded or modeled’ . They rearrange bone and flesh, molding it in a plastic manner.

Wikipedia says, “This meaning [of plastic] in English is seen as early as 1598.The surgical definition of “plastic” first appeared in 1839, preceding the modern “engineering material made from petroleum” sense by 70 years.”

Nope. “Plastic” stems from an old Greek word basically meaning “moldable/malleable”. That’s the context in which the term is used when we say “plastic surgery”.

^^ Agree. When people use the term “plasticity,” they mean a quality of something that can be shaped or molded, in either a literal or a figurative sense, and not something made of the same material as baby toys and kitchen storage containers.

Silicon is considered a plastic.

While the term does descend from the idea of plasticity, or able to be shaped, a number of types of modern plastics are lightweight, extremely strong, and more easily accepted by the body internally than metals. It is also possible to provide any shape necessary.
So, while plastic is not used to replace skin, it does make a very good material to provide desired shaping.