Um, I'm just wondering if mother-in-law maybe has more of an issue with you, and your dog is a handy opportunity to play that out? Sounds very passive-aggressive to me.

Hope that's not the case, for your sake.
When somebody disses your dog it feels the same as someone doing that to your child. It's personal and it hurts. I call them out on it, but that's me.
Once, a neighbor made a comment referring to my yorkie as a fancy rat. This man had an awful hairpiece. So I said, "Rat? Maybe, kinda like the one on your head, only much cuter! "