Possibly, but it depends...
The beverage entitlement is a standard fountain drink such as soda, coffee, tea, iced tea, lemonade, but some of the Disney owned TS restaurants (e.g. 50's Prime Time, Whispering Canyon Cafe and Sci Fi) will let adults order a milkshake as their beverage, but you can ask at other TS restaurants. Likewise it can usually be ordered as a dessert. At some places you can have one for your beverage and another for dessert.
Unless it is on a children's menu, it cannot be ordered as the child's beverage or dessert using the child's credit, but an adult can use their own entitlement and share with a child.
Also, third party restaurants will not allow it as an adult's beverage because they are reimbursed by Disney according to the rules of the dining plan (standard fountain beverages only). If the milk shake is on the dessert section of the menu, it can be ordered for dessert.