It has a very similar meaning, but I don't think it's paedagogical to accept translations that are unnecessarily far from a more direct translation. The point of these exercises isn't just to get the meaning of the sentences, but to parse out why the words are in this or that order, and how they influence each other.
As a retired teacher of several languages, I actually agree with your point about not necessarily having to accept translations that differ too much from the structure or wording of the original, depending on what it is that one is trying to teach. However, I don't agree with on your assessment of my alternative. And I do think it's important to capture the point or essential message of the original. Certainly several others here have recognized that уже may be rendered as 'already,' and not just 'no longer,' depending on the rest of the English sentence. And as a native speaker of English, I don't think that 'She already has no milk,' captures the sense that this is a potential problem as well as 'She is already out of milk' or Duo's 'She no longer has milk.' For me the point with уже here is that there was a change of state, from having something to not having it, that has implications. To 'be out of something (already)' is a very natural and valid way to express that.
It's just an idiom - уже + a negative construction means "no longer".
У неё уже есть молоко - she already has milk (because she went to the store earlier) У неё уже нет молокa - she no longer has milk (because she drank it all)
Both indicate that a transition from one state to another has happened, but it's probably better to just memorize them separately - there's no exact correspondence to English.
Correct solution: • She has no milk already. • She no longer has milk.
Say what now?
It sort of means "already" here too.
"already not" makes sense, if you think about it. Think of it like "It is already the case that (something) not (something)", which is valid English. In Russian and Spanish for example, it is just the case that this "already not" is the normal way of speaking. In English, it just happens that "already not", although logical, it not used, and "no longer" or "not ... anymore" is the way the sentence is built instead.