Both are true. A “plastic”, used as a noun, refers to a class of moldable polymers. Other substances which are malleable and can be molded into solid objects are called “plastic”, used as an adjective.
Plastic certainly isn’t “a” material. It can be any one of a very large class of materials, or even a mixture of multiple such materials (ABS, for instance, which Lego and some 3d-printed objects are made from, is a mixture of three different plastics).
transparent (clear) plastics like polystyrene and polycarbonate tend to be rigid and more brittle, and more prone to cracking. since the album art/booklet was meant to slide in to the CD case cover, the rigidity clarity of the plastic was more important.
kids toys and the other examples you mentioned are made from different polymers. “knock-around” stuff like big wheel trikes and the like are generally made from polyethylene (PE,) which is flexible and resistant to impacts. but PE is typically not clear, at best it’s translucent (milk jugs.) a PE blend which is transparent (PET) is used for soda bottles, but it’s too flexible/bendy for a CD case cover. ABS and blends like PC-ABS tend to have good impact resistance, but again are almost never transparent. there is ABS out there which is transparent, but guess what? it has much less impact resistance.
(Note: polycarbonate is what the CDs themselves are made out of. Much stronger in this case.)
I was thinking about the history of media packaging. I think audio cassettes were the first to use polystyrene shells. I don’t think people generally kept 8-tracks in individual plastic shells.
The shells of cassettes have an advantage over CD jewel cases in terms of compact size, wider edges, etc. All of which reduce the forces on and strengthen the hinges. (But they still could be broken with little effort.)
I think the experience with cassette shells lead to the adoption of CD jewel cases. Note that CDs were initially often sold in long box packaging. That increased the total cost of packaging so keeping the jewel case part cheap had to be a factor.
(One oddity about the heyday of cassettes vs. CDs was that CDs were much cheaper to produce but carried a premium price.)
Video cassettes eschewed clear cases in favor of cardboard and soft plastic packaging.
Of course the successor to BluRays will have no packaging at all.
I know the OP didn’t ask, but: after having years of experience mailing CDs I have found a design that withstands the USPS machinery: the clamshell type, such as these (not these in particular, just representative):