Mava, this brings a new wrinkle to the issue. When the OP was describing plastic, I was thinking hard plastic not rubber coated. I now wonder if the rubber coating had been failing all along and developed a non sticky skin. When the car was cleaned, this skin was washed off leaving the previously hidden sticky mess.

So, if this is the case, the cleaning shop isn't liable as they had no way of knowing that cleaning would cause a problem. If the owner had cleaned the plastic with Windex they might have had the same problem. The only way to be sure is to post on more message boards asking if others have similar problems.