The purpose of a dog tax was to create a fund that could be used by the town to reimburse citizens for any damage done to their livestock by dogs. Such a tax was first implemented in Windham in 1866. In 1888, the dog tax raised $69.00; it seems that none of this tax money was paid out for its intended purpose. When there was leftover dog tax money in the town treasury, the remaining money was used to fund education in the schools of Windham. Eventually this dog tax was phased out and replaced with the more modern dog license. Unlike the dog tax, the purpose of the dog license was more to accurately assess the number of dogs in Windham's households, and provide a way to locate the owner of lost dogs. These licenses have, and are today, issued as tag to be placed on a dog collars; note the 1950s dog license tag shown above.
Sources:
http://www.thedogtrainingsecret.com/blog/history-dogs-america/
https://www.genealogytoday.com/articles/reader.mv?ID=974