On the Dream & Fantasy, there is a real fridge in the room. On the Wonder & Magic they are "cooling boxes". If you have a medical need, you can request a real fridge on the Wonder & Magic for your room.
One note, none of the the fridges have a freezer compartment. And
DCL most likely will not freeze an ice pack for your personal use. If you need something like this you might look into those first aid snap-paks that freeze up when you break the vial in them.
From the DCL site:
Beverage Policy
QuestionWhat is the policy regarding bringing alcoholic beverages on board?
Answer
Alcohol is permitted on board. However, any fragile items (including alcoholic beverages) must be hand-carried on board in the carry-on of an adult Guest age 21 or older (dimensions can't exceed 22" wide, 14" high and 9" deep).
Alcohol brought on board may not be consumed in any lounge or public area.
Guests who arrive in the dining room, including Palo and Remy, with a bottle of champagne or wine will be charged a corking fee of $20.00 per bottle.
Please be advised: Disney Cruise Line reserves the right to remove fragile items (including beverages) from checked luggage. In such cases, items will be stored and returned at the end of the voyage. Beverage containers are considered fragile and have previously caused damage in checked luggage. All fragile items must be transported inside carry-on luggage.