"Azure" is an English color-word, but an English speaker recognizes "azure" as a kind of "blue". But "blue" is not a kind of "green" or a kind of "purple" or a kind of anything else. Hence, "blue" is a basic English color-word.
Typical English dialects have eleven basic color words: white, gray, black, brown, pink, red, orange, yellow, green, blue, and purple. Other color-words in English are considered variants of one of these eleven terms. That wasn't always true ("pink" is the most recent addition, dating to the early 1700s), but it's pretty well-established by modern English speakers. Other languages may make fewer distinctions (e.g. until about the last century Japanese used ao(i) as a basic color word that encompassed both blue and green, so e.g. a traffic light is aoi by tradition even though the modern word for green is midori) or more (e.g. Russian distinguishes light blue goluboi and dark blue sinii similarly to how English distinguishes red and pink). To an old Japanese speaker, blue and green were both shades of ao, and to a modern Russian speaker, goluboi is as distinct from sinii as "red" is from "pink" in English (they are not both "shades of blue" any more than pink and red are "shades of reddish" in English).