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Mister Muffykins is for purse dogs. IDK if we have a broader trope for other animals, but Distinctive Appearances might cover the similarities between bag-size/pouch-size pets.
Your goateed philistine is sashaying towards us. | 🧱I read that page while I was searching for the correct trope, in the example I need this for, that trope does not apply. My interpretation is that page is about annoying little dogs, not pets in fabric carriers. The dog in question is that small, but not annoying, (except to the villains of the show).
Also, I was thinking another example could be Rufus from Kim Possible, who is not a dog.

I couldn't find a trope where a pet is small enough to be carried in a bag of some kind.
Usually a purse dog, but the example I saw is carried in a waist pouch.