What properties allow something to be considered "milk"?
I will disagree with Johanna here. While hers sounds like a reasonable definition, it is not how the word is used in practice. 

Milk is 


A) Cow's (also goat's, sheep's, camel's and mare's) lactated fluid, or 
B) Any liquid which kinda looks like A), doesn't have an overly strong taste, and there is a convention of being called a milk. It can in many cases be a substitute for A, including when drinking straight out of the mug. 


So not only are there nut milks like almond milk which contain both fat and carbohydrates, but there are also grain milks like rice milk and oats milk, which have practically no fat and get their white color from the starch in the grains. This goes against the "fat in water emulsion" definition, but it is common usage. 

There are also foods which have "milk" in their name while nobody would say that they are a milk, like the "donkey milk" wine. 

In the end, there is no rule. Whatever people call "milk" is a milk. And they call "milk" whatever reminds them of the milk they have encountered before.