We to stick to the subject. If it's HER drinking, presume it's her milk. Unless the sentence specifically mentions a man being in the picture don't assume it's his milk. And vice-versa. For example:
"Le homme boit son biére froid." A or B?
A) The man is drinking her cold beer. B) The man is drinking his cold beer. . . .
I have some serious doubts about the validity of this English translation. In the phrase, "cold milk", "cold" is clearly an adjective describing the temperature of the milk. However, in the sentence, "The girl is drinking her milk cold", "cold" acts more as an adverb describing how the girl drinks her milk.
This may be a distinction without a difference, but it is definitely a difference. The position of "cold" changes the gist of the sentence.
For example, "The girl drinks her milk cold, and so she refuses to drink this warm milk."