Actually it is plastic--it is acetate over fiber glass fabric or other fabric.. In fact the earliest versions were made of old discarded movie film and burlap. Anything that is activated by heat or, esp., solved in acetone is nearly by definition plastic.
It is also crushable and unable...short of extraordinary measures...to be restored.
Celastic has good elasticity while being stiff. Heat activated puffs are a lot more bouncy.
Neither behaves like those plastic bucket that gets brittle under the sun that you used as an example.