The term seems to indicate an informal society that is controlled by its members and is not subject to outside regulation. I found a reference to Richard Nixon being in a Republican one in 1949. Another site has a Historical Preservation group saying "That they were serious preservationists not just some "Chowder and Marching Society" This quote refers to Tammany Hall in those terms though I realize the quote is not from that time:In the beginning was Aaron Burr, America's first professional politician, who in 1800 taught a chowder and marching society named Tammany Hall how to assemble lists of voters and canvass them, asking, with special urgency, if they were willing to give money. It worked so well, Burr took New York City away from a stunned Alexander Hamilton, the man who maintained that the rich had to have a voice in government.
...I would assume that "Marching Society" would refer to the Irish Proddie groups who march in the "Marching Season",
to remind others of the victory of King Billy in the
Battle of the Boyne. Not much help I know....