While the word “napkin” apparently came from the French, the French themselves abandoned it in exchange for the much classier sounding “serviette.” This was a good decision as far as I’m concerned , as serviette is a much sexier word than the flatfooted napkin.
Interestingly, though I’ve almost always seen this instance of the word “napkin” here as a synonym for handkerchief, the etymology site does not mention the word’s period as a handkerchief.
Shakespeare seems to almost always use napkin in this sense. Even the very most famous handkerchief in the canon is called a napkin. (I’m talking about Desdemona’s “napkin” here.)
One of the origins of the word relates to the material so that it basically means little linen. (“Kin” being a diminiutive.) I wonder if, in Shakespeare’s time, a napkin like you use at dinner was actually the same as a handkerchief. Like, were they both just little squares of linen?