Emerald is the May birthstone, and this is a good place to state plainly what the modern birthstone list is.
It is a trade standard from 1912. The National Association of Jewelers, the American body now called Jewelers of America, met at its annual convention in Kansas City in August 1912 and adopted an official list of birthstones. That list is the direct ancestor of the one printed on jewellers' charts today. It has been revised since, by committee, for commercial reasons: the Jewelry Industry Council of America added alexandrite for June, citrine for November and pink tourmaline for October in 1952 and replaced December's lapis lazuli with zircon; the American Gem Trade Association added tanzanite to December in 2002; and spinel was added to August in 2016. Britain's National Association of Goldsmiths issued its own separate list in 1937.
A scheme that acquires a new stone whenever a supplier has one to sell is a marketing instrument. That is not a hostile characterisation; it is what the revision history shows, and one critic quoted in the standard reference literature describes the 1912 list as nothing but a piece of unfounded salesmanship.
What makes this worth insisting on is that the 1912 list is almost always presented as ancient. It is not. There are genuinely old stone-and-month and stone-and-sign schemes, and they are different lists, they descend from attempts to correlate the twelve stones of the priestly breastplate with the twelve months and the twelve zodiacal signs, an idea found in Josephus in the first century and in Jerome in the fourth. Those are real antecedents and they deserve to be cited as such. The stones on the modern chart are not the stones on theirs.