Some months are associated with multiple birthstones because there is both a traditional and a modern list of birthstones.
The traditional list of birthstones originated in Poland between the 16th and 18th centuries. In contrast, the National Association of Jewelers released the modern list in 1912. The National Association of Jewelers created the modern list to suit the contemporary taste, and this is the list most jewelers go by today.
While most months have just one birthstone, some months have multiple if you consider these different lists. December, for example, has both blue topaz and turquoise as its birthstones. August has peridot, though many people consider spinel as an additional birthstone for this month.