The answer mentioned here is correct. The measure of cosine distance as a measure of similarity only makes sense under some specific assumptions.
- That it is possible to represent multiple complex objects as commensurable entities.
- We can use quantitative methods to find qualitative answers. For example, we can measure the similarity using some numbers.
One can argue that these assumptions can be used as a guiding principle for a wide range of measures. Indeed they are. But cosine similarity is more popular because they match well with our spatial intuition and common sense. If you ask someone totally unrelated, to state the similarity of two documents between -1 and 1, then the answer would probably close to what cosine similarity would give as well.
There are other factors that also make cosine similarity a good choice. When thinking about the similarity of two documents, we do not care about the order of the words or other specific grammatical constructs. Hence the cosine distance, you will notice, does not take into account these factors and hence kind of captures that essence.
Lastly another advantage of cosine similarity is that, in high dimensional spaces such as text word embedding, are essentially non-intuitive to humans due to our inability to grasp non-euclidean spaces. Hence to understand meaning in a non euclidean space we need some way of mapping this high dimension to a low dimension space. Thus cosine distance helps the researcher to visualise the vector space to a large extent.
So to answer your question, the absolute value of cosine distance does not make sense by itself. It only makes sense if you are comparing between multiple choices. If you started with "king", then chances are that the text is talking about a "man" rather than a "woman" or a "bird" based on the cosine distances. Just the cosine distance between "man" and "king" has not value by itself.