The history of the song's origin is clearly English, with Irish, American, and music hall chorus variations added with time. I don't recall anyone claiming that Patrick Flemming was American.

Jar, in the 19th c was a common name for the container in which wine and liquor, including whiskey, was sold. More recent usage is jug. Medieval French jarre, Spanish jarra (from Arabic), was applied to an earthenware drinking vessel. Original usage in England (see OED) was for a large earthen vessel for holding water, oil, wine, etc. In 1732, Pope wrote of a Spanish vessel bringing 1000 jars of oil.
Our modern canning jars are the result of studies by the late 18th c. Frenchman, Nicholas Appert and have nothing to do with containers for liquor, oil, etc.

"Jar" has nothing to do with the song's origin, one way or t'other.