Seems the tarrier of song may have arrived at about the same time as the terrier breed. The spellings were interchangeable for man and dog.

Wiki: “The first breed club was set up in Dublin in 1879. Irish Terriers were the first members of the terrier group to be recognized by the English Kennel Club as a native Irish Breed – this happened just before the end of the 19th century. The first Irish Terriers were taken to the US in the late nineteenth century and quickly became somewhat popular.”

Just before the song and the play came out “Honest John” Kelly appeared as a "Tammany Tarrier" in the Puck political cartoon Our national dog-show (Artist: Bernhard Gillam)(N.Y.: Keppler & Schwarzmann, 9 May 1883.)