In addition to other comments it does contain cold. Capsaicin interacts with the temperature sensors on your tongue, so anything cold will provide temporary relief. Most folks store their milk in the refrigerator but might not have fridge cold water or other drinks.
Also, capsaicin is basic and milk is slightly acidic. This means milk can have a small effect on the shape of capsaicin protein molecules this changing its properties. As milk is very slightly acidic this is not a strong effect, but it explains why other more acidic foods are paired with spice (beer, soda, margaritas).
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