Plastics are awesome. One of their characteristics is that they easily degrade in UV / sunlight. In my experience that is the weakest link in its usage. With that said, there are a whole lot of plastics used on modern cars.
Generally headlamp lenses are made of polycarbonate. The polycarbonate material has a thin coating sprayed on top of the lens to help provide extra UV protection.
Red tail lamps lenses are virtually all acrylic. It's got decent properties, molds well and passes light consistently. And plexiglas is basically Acrylic.
As for why plexiglass is not used in cars? My guess is you really need to follow the dollar bill $$$$$$$$. Glass forming dies aren't nearly as expensive as molds necessary for things like headlamp lenses. The molds (Dies) for these products is crazy expensive. You can't imagine.
I did a whole lot of engineering for a US based Original Equipment Manufacturer (OEM) for both exterior glass and exterior lighting products. I will say the molds for polycarbonate headlamp lenses were some of the most highly engineered (and super expensive) tooling we had to deal with. Because of the size of the molds you really can only have one person working on a mold at a time. Its not possible to put ten people on the job and have it go ten times faster. That steel is super expensive grade, super hard. If you do something wrong or try to go too fast you can crack a mold.
Some of those tools can take up to year to produce from start of design till first parts come out of the press. From my experience, headlamp design and tooling is on the critical path for a vehicle's development cycle.
No way you'd want that hassle or tooling investment cost for door glass or windshields or rear window. Float glass is still expensive if you had to build up a glass factory from scratch, but once the factory is built glass is very reasonably priced. Heck, the raw material for glass is essentially sand. Float glass has been around for a long time. True you can only shape glass as a ruled surface but designers have worked with that for a long time.
And glass doesn't degrade in UV light. Not one bit.
For the question "Why did the manufacturer do that?" the answer is often.. because its cheaper that way. (Follow the money!)