That mother is out of touch with reality. Seriously...overnight (cause she saw the poster at the vet the next day) she decided it was their dog and went out and bought stuff for it? She needs a meeting with the Cluebat. As in WHAP!! Get a clue!!! Then she has the massive brain misfiring of not sitting down with her kids and saying "Hey listen, I know you really like this dog but there is a very sad family out there who misses him, and he really belongs to them. We will go to the pound and find you a new dog, but we have to give this one back, go say your goodbyes and daddy will take you to ice cream while the family comes and picks up her dog" Even worse she let them be in the room with Maverick when he was going on. I'd call the mother, tell her I want my dog back, and after I had pried him from the fingers of the kids who will one day be devastated by the word "No", i'd tell her exactly what I thought of her...well...out of earshot of the kids though.
I recall many tv sitcoms where they faced this exact thing, the kids found a dog, fell in love (difference being the parents immediatly informed the children it was prolly lost and someone mised it) and when the real owners came the kids had to give the dog back. Granted they usually ended up with a puppy after that but still...even HOLLYWOOD knew it wasn't right to let the kids keep the dog.