In my family, we usually said "we're out of milk". But I have also heard "the milk is finished". In the past tense, you could say "I didn't finish the milk. It was already finished when I got home". I guess you could say "The milk has been finished for a couple of days. Go buy some more already" but that's rude. "The milk has finished" is just ridiculous.
Several people have reported for the last 2 years that "The milk has finished" would never be said in English. I'm flagging it again for everybody's sake. I might say "The milk is all gone." It's also likely to be said in the other direction, such as "Someone finished off the milk; it's all gone." There are other situations, such as "The milk has finished turning sour. Go ahead and make pancakes with it." or "The milk has finished coming to a boil. Turn off the burner." Dec. 2019