If your fluffy little puppy is perpetually begging you for a taste of whatever it is that you're having at the moment, remember that it's never a smart or safe idea to feed him human food or drink. Milkshakes, which are dairy products, are certainly no exception.
For the first several weeks of a puppy's life, not only is it important not to allow him to drink milkshakes, he actually shouldn't be consuming anything at all other than his mother's milk. Mama fulfills all of the nutritional and hydration needs for nursing puppies, and that's all they should get. If for some reason the mother isn't involved in rearing a puppy, you can take over for her by feeding puppy formula with a commercial milk replacer. Puppies begin a gradual weaning process around 3 weeks of age, when they start sampling solid food. Weaning continues often to about 8 weeks of age.
Even once a puppy is past the crucial nursing weeks, milkshakes are a bad, bad idea, and apt to make him miserable. The main ingredients of a typical milkshake are ice cream and cow's milk. Milk and milk-based products can trigger diarrhea and severe abdominal pain in a puppy or a dog, because they can't digest lactose. You do your puppy or dog no favors when you let him eat any type of dairy product. A milkshake also is full of sugar, and sugary desserts do nothing to satisfy the nutritional requirements of growing canines.
Milkshakes aren't the only "human food" nuisance to wee puppies. Never allow a puppy to eat anything other than food that is labelled as specifically being for puppies. The little guys have different nutritional needs than adults, and thrive best on foods formulated for fast-growing puppies rather than regular adult dog food for the first six months to a year of their lives. The ASPCA lists some people foods as toxic to dogs. Among them are macadamia nuts, chocolate, grapes, garlic, onions and avocado. Many other foods are on the list. Your safest bet is to feed your little one nothing other than puppy food.
The only suitable drink for a dog is fresh, clean, cool water. Nothing else. Water is essential to proper canine nourishment. Keep clean water within your little buddy's easy reach at all times.