I'm afraid so. Cats can digest milk, but dogs can't. I can't remember the exact problems caused by dairy, but I can look it up, if you'd like? I think I will, just to appease my own curiosity. Will post it in a few. But I'm afraid I would advise against letting your dog having the cereal milk. I'm sure he won't understand when she stops giving it to him, but perhaps you could provide a substitute for a while, then slowly wean him off that as well, to teach him not to expect anything after your wife finishes her cereal?
Milk is not healthy for adult dogs as they are lactose intolerant. They can't absorb this milk protein and consequently suffer diarrhea. Give your dog plenty of water and only small amounts of milk.
A dog can develop a sweet tooth too. Play it safe by never giving your dog sweets and chocolates. They promote tooth decay and gum disease. Remember that dogs don't get to brush their teeth. Chocolates are especially toxic to dogs as they contain theobromine. When a dog eats a large enough quantity of theobromine, toxicity signs appear. The dog will then vomit, pant, suffer diarrhea, muscle tremors, restlessness and incontinence. Medical attention is necessary to save its life.
There's more, but this is what I found on milk and milk products. Ice cream is richer than milk, of course, and is definitely not recommended. (Especially since it also contains sugar.) I remember that much from a variety of other sources.
Sometimes it may take a while for the build-up to occur in the digestive system, so you may not see the ill effects right away. I wouldn't chance it, myself. I give my Piper and Wilson probiotics every now and then, to counter any ill effects they may have to any milk by-products they might have digested. (Off the occasional pizza crust that might have cheese on it, or when one of the little buggers swipes people food from the counter! Bassets are ntorious counter cruisers, and Piper is an active little sneak! *G*)
Check Doctors Foster and Smith for the probiotics and other supplements. Bassets tend to need more Omega oils than other dogs, so I use Foster and Smith's Omega supplement on their food, which makes their coats super shiny! I also use their toothpaste, which is an exercise in patience that is unmatched! *VBG*
Hope this helps!!