Can't verify his historical citations, but @Fumduck seems mostly right for certain reasons. There are a few reasons to shake a cocktail. You might want to change the color of a cocktail. Easiest way to test this is with a margarita. Just the simple act of incorporating a lot of air into a margarita will change the color a little bit lighter, and change the mouthfeel to something slightly more airy as well. Anything with any sort of cream or egg whites in it, it helps it emulsify, and turn even lighter in terms of mouthful. But if you every make a martini, never fucking shake that shit. People want their white liquors looked clear, like an albino Scandinavian looking through a window. You shake that shit, it turns cloudy. No significant taste difference from what I can tell, but people care.