That's a pet-peeve in general, but it fits in with the "furbabies" and "my pets are my children" thing taken to an extreme. I don't know if the reading is off-base, because it is so outlandish, but it's not uncommon rhetoric among pet owners even beyond the adoption parallels.

It drives me nuts that my mom calls me the sister to her dog. The dog doesn't care when mom asks her "do you wanna play with your sister?", but it makes me uncomfortable and she won't listen to me when I tell her that I am not sister to a dog. Of course I am, the dog is her child, and so am I. Ergo, we're sisters.