To me, this is just a ploy to get closer to you. It's a somewhat passive aggressive route back into your lives. As in, she doesn't want to bother YOU. She doesn't want to cross YOUR boundaries, she just wants to talk to the dog!
My SIL started the same tactics with our dogs, which she called her "nephdog" and "niecedog." Due to other boundary issues, we didn't want to spend a lot of time with her on the phone or have her visit at our house. So she took to calling to "talk to the dogs" or she would drop by for a visit "with the dogs."
We just chalked it up 1) her being a very enthusiastic pet owner and 2) trying to work around the boundaries we'd established. We refused to put the dogs on the phone, by saying it wasn't possible, then told her we had to hang up. Or we would stop her at the door and tell her that it wasn't a good time to visit. Eventually, she dropped that tactic.
The only time is got really troublesome was after we had our daughter. (The first grandkid in the family.) SIL felt displaced, so she started demanding "equal treatment" for her dogs, since they were my in-laws' "granddogs." If my daughter got a special treat, SIL would say, in the dogs' voices, that the "granddogs wanted a treat, too!" If my daughter got a little "just because" gift, SIL would speak for the dogs saying that their feelings were hurt and wanted a gift, too. If the dogs didn't get Christmas presents... it got ugly.