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.

Note there are specific European Union regulations that define the terms "milk" and "dairy". This article from the Food Standards Agency (UK) summarises this: Guidance On Legislation On The Protection Of Definitions And Designations In Respect Of Milk And Milk Products

Specifically it says:

The terms ‘dairy’ and ‘milk and milk products’ are used synonymously and are used to describe products from a normal mammary secretion obtained from one or more milkings.

There are a number of other protected terms, eg "cheese", "butter", "yoghurt". These can only be used for products made from dairy milk.

So in the EU, plant-based milks can not actually be labelled as such. Instead, soya milk is labelled as something like "soya drink" or "soya alternative to milk" or just "soya". Soya yoghurt may be called something like "soya dessert"

Milk is a stable emulsion of fat in water, basically tiny drops of fat suspended in water. To qualify as milk, rather than cream, or whipping cream, it has a certain fat percentage, usually less than 3%. This is why really low fat milk tastes like water: it is basically water. Such a stable emulsion can be produced in lots of ways, including by cows or goats (or any mammal), extracted from soy or almonds, or come from coconuts. Depending on where the milk comes from, it will taste different, but since the basic ingredients (fat and water) are the same, they can usually be substituted for each other.

So, to answer your question: there are plenty of white liquids produced by animals or plants that are not milk (several plants have white sap for example). The essence of milk is that it is an emulsion of fat and water.