From the early sixteenth century, in England, a popular nickname for anyone with the name Joan or Joanna was Jug. Especially was she called by that name if she were a servant or of unattractive appearance.
No one knows why, any more than one can explain why the name “John” became altered to “Jack” as a nickname, or “Margaret” to “Peg,” or “Mary” to “Molly” or “Polly.”
Be that as it may, it was during the same century that potters began to turn an earthenware vessel somewhat similar to, but larger than, a pitcher.
Perhaps the squat shape of these early vessels reminded some wag of a popular barmaid named Jug who was of similar shape; or perhaps it was just because other drinking vessels were already known as jack and from the nicknames of John and Gillian.
The fact cannot be determined. The full name, Joan, which would be Jeanne in French, suggests the possibility that “Dame Jeanne” (demijohn) might have been a French translation of jug.