There’s the story that Ruchama Shain wrote in her book about her father, “All for the Boss” when she got a beautiful doll as a child, and her father smashed its face so as to avoid the prohibition of having a pesel (idol).
So there could be an inyan about dolls being a graven images and not allowed (when whole). I don’t know if that would only apply to a porcelain doll, or to any doll, whether made of plastic or soft material.
I actually never heard anyone else having this practice. I have seen dolls of all types in many frum homes, and they are intact until the kid over-uses them.
If you want the parent to still like you, avoid the dolls that make noise or sing the same song over and over and over again.