By plugging "official song" and "city" into www.google.com, I came up with these: (no doubt there are more)

Cedar Rapids, IA
Fall River, MA
Franklin, PA
Koshigaya City, Japan
Livermore, CA
New York, NY
Richton Park, IL
San Francisco, CA
Schenectady, NY
Spokane, WA
Union City, CA
Washington, DC
Wellington, New Zealand

Halifax, NS seems to be the only place that has a deliberately funny official song. It is sung to the tune of Mussorgsky's "Pictures at an Exhibition"!

I would expect that, in some small towns that only have one high school, the school song becomes, by default, the semi-official song of the town. As an example, see Ponca City High School, Ponca City, OK. Even the words to the song are ambiguous: "Ponca City, we love you." Does that refer to the city or the high school? It probably doesn't matter to the residents of Ponca City -- if you love one, you love the other. In my limited experience of small towns, the people identify closely with the high school. The old-timers and all their friends went there. Their kids go there. Their teams are the only teams to cheer for. If a team wins a championship, they announce it on the big sign that says, "Welcome to ----, home of the -----, 19-- state champions!"