I have several cows on the Isle of Awakening, but I can't figure out how to milk them. I know I was able to once before, 'cause I have milk, but I can't figure out how I did it. I tried allowing them to have a baby cow in case that was necessary for them to produce milk, but I just can't collect any. Can the cows be "milked-out"? Like they only give milk once?
In order to obtain milk from a cow, you have to have your water jug (the bottomless pot) equipped; approaching a cow will then cause a "milk" interaction prompt to appear in addition to the usual "stroke" interaction prompt that creates milk you can collect. (Note that it doesn't put milk in the pot--it just appears in the world as its own object once you use the pot on the cow. It's a bit of an unintuitive system.)
There's a video here showing the pot-equipping method of milk collection, in case that helps you as well.
As far as I'm aware, cows cannot run out of milk permanently.