Getting Milk in Nintendogs? Not a Feature

Milk does not exist in Nintendogs (DS).

Your dog consumes:

• Dry Food / Canned Food (Bought at Shop).
• Treats: Biscuits, Jerky, Cookies, Ice Cream (Shop / Contests).
• Water: Auto-refill bowl (Tap water icon).

No milking, no milk items, no lactation mechanics. You may be thinking of Harvest Moon / Story of Seasons / Stardew Valley (Cow/Goat milking) or Animal Crossing (Milk bottle furniture/item).