IIN I cringe when somebody refers to a pet as their "fur baby"

That's animal cruelty in my book.

There are a few cats that are so chill that they wouldn't mind being taken out like that, but domestic cats are instinctively solitary territorial animals, and hauling them to new places is generally pretty stressful for them.

This is something the woman was doing for her benefit, not the cat's, and therefore it's messed up.