A metric (https://en.wikipedia.org/wiki/Metric_space) or a distance (as in premetric) takes two elements of a set and maps the pair to a real number (or maybe even a complex number). It also might have a couple of further properties that must hold for the respective function to be a metric.
A measure (https://en.wikipedia.org/wiki/Measure_(mathematics)) on the other hand takes elements of a set's $\sigma$-algebra and maps those to a real number (or maybe even a complex number).
Let us assume that $X$ is the space over which I define my distance.
Then, usually, the question "What is the distance between the three points $a, b, c \in X$?" does not make any sense, as a distance is only defined for a pair of elements of $X$.
The $\sigma$-algebra of $X$ must include the complement of each of the $\sigma$-algebra's elements; thus, the $\sigma$-algebra of $X$ must include elements (sets) that are not pairs, but include more than two elements of $X$.
As a result, it seems to me, that a distance is not a measure. This is bewildering to me, as I thought that "measure" is a generalization of "distance" (and other concepts).
What am I missing?
EDIT:
The misunderstanding might come from the number of times that "measure" (verb) is used in the Wikipedia text for "metric". E.g.: "The distance is measured by a function called a metric or distance function."
Maybe I should think about this like so:
If one says "the distance is measuring the length of the line", this actually be translated like to:
There is a line, which has the end points A and B. We want to measure the measure "length" of the line, which is a measure. While the line does have a length, it is difficult to measure it directly. However, we can simply calculate the distance between the points A and B (not the "distance of the line") and use that distance as a means to calculate the length of the line. The line has a length, its end-points A and B have a distance.
This "translation" is by me.
Best Answer
"Measure" is meant as a measure of "volume", or "space", how much "volume" a set takes up in an ambient space. Thus, nothing to do with distance.
Keep in mind that the names given to precise concepts in mathematics are usually chosen to remind you of some intuitive concept related to the precise definition, but this intuition can also mislead you if, for example, the intuitive meaning of the word is too vague or there are multiple meanings for that word in natural language. That's why we have precise definitions in the first place. Thus you shouldn't base yourself on the word itself to define the concept, the word should just be a mnemonic.