A discussion of this question caught TPP's attention while driving across town the other day. It basically involved someone representing the dairy industry, and people who produced/consumed "plant milks", soy milk, almond milk, etc. (it's quite a list). Milk/milky is one of those not very precise words that some people are trying to narrowly define usually because of an economic interest in doing so. Milk in the dairy sense is a secretion of mammary glands, and while primarily cow, goat, camel, and others also produce milk, as do the rest of mammals, and to some extent monotremes as well.
These milks are emulsions, complex mixtures of organics suspended in water. And generally they are milky, a somewhat whitish, liquid. The word even gets used in astronomy; galaxia refers to milk and our galaxy, also referred to as the milky way, is because the disk of distant stars across our sky looks "milky" to the naked eye. As a side note, a small neighboring galaxy is called Snickers because next to the Milky Way it's just peanuts.
Now the plant milks are basically an emulsion made by grinding up seeds and extracting the cell contents to make a milky emulsion. Although the liquid endosperm of a coconut (big seed), the coconut milk, is a similar but self-made emulsion.
This still leaves another category not mentioned or even recognized by the people in this program. Lots of plants probably 7-8% produce some form of latex, a milky liquid, that in some species contains elastic and non-elastic molecules that can make gums and rubbers. The plants that secrete such latexes are sometimes so labeled, like milkweed. This is what the botanist thought of when asked what milk is, something secreted by laticifers, specialized cells or ducts. These are usually considered protective secretions to prevent or limit herbivory. Raw opioids are obtained from a gummy latex secreted by injured poppy capsules.
So sorry dairy people you do not get to co-opt the word milk and prevent everyone else from using it (their actual mission). It's just too general a descriptive term.
Camponotus: A Sugary High
1 hour ago in Catalogue of Organisms