That's funny, my first "real" bass was also an '81 CAR ... but it was an Ibanez Blazer that in hindsight was clearly imitating that Fender CAR.
It looked like this (not mine) - note the brass knobs, white pickguard, brass bridge with heavy tailpiece... sure there are obvious differences, like the all-maple neck and the body shape, but I have to think they were consciously imitating the Fender, even the bridge design (brass, and fatter in the back than a usual P bridge). And I unknowingly took mine closer to the Fender look for a while when I had a creme DiMarzio Model P in there!
I still have my Blazer in fact, though it has been mutilated beyond recognition since then... see my avatar!