Why is a game between local teams called a "derby"?

The origin of the term "derby" to describe a game between local teams is unknown. According to Wikipedia,

"The phrase may have originated from an all-in football game (Royal Shrovetide Football) contested annually between the two halves of the English town of Ashbourne, Derbyshire... Another theory is that it originated from The Derby, a horse race in England, founded by the 12th Earl of Derby in 1780... Another widely reported [theory]... is that the phrase came about from the city of Derby itself."

More info: https://en.wikipedia.org/wiki/Local_derby

As often happens, Wikipedia has no source for its speculation. Mark Smith, a billed "expert" on the allexperts.com site, opined in favor of the connection to the Shrovetide game in Ashbourne, Derbyshire, but also offers no support:

en.allexperts.com/q/Professional-Soccer-European-1848/Derby-Games.htm website no longer exists

The link to The Derby on Wiki seems spurious. It clearly seems to be tied to Shrove Tuesday town football, although whether in Derby itself or just in Derbyshire is the question that needs a definitive answer.

Without firm evidence, often all we can do in etymology is speculate.
Since the 18th century, when the famous horse race at Epsom was named the Derby, various contests have clearly taken the name, from the similar Kentucky Derby to soap-box and demolition derbies.
"Derby" as used in the question is surely just a contraction of "local derby", i.e. a contest between local teams.

There is a source that agrees with your claim, almost word for word:

The Derby is the name of the premier horse race in England and the contest was founded by the twelfth Earl of Derby in 1780. Since at least as early as 1840 'derby' has been used as a noun in English to denote any kind of sporting contest. A local derby is just a sporting contest between local rivals. Early uses of the phrase referred to football (soccer) matches and the first printed citation appears to be the Daily Express, October 1914:
"A local Derby between Liverpool and Everton".
Such matches between the two clubs are still energetically contested. Many other clubs in this and other sports now meet to contest their own 'local derby'.

http://www.phrases.org.uk/meanings/235375.html

Other sources disagree:

Hundreds of years ago, it was often at holiday times when the citizens went out onto the streets to celebrate. But troubles began when neighbouring parishes or regions of the city would meet. Brawls became common place. Eventually, as folks became more civil towards each other (as Christianity and other religions began to flourish), the "brawls" became more organised. Ball games would ensue. Some kind of football was the most popular event. Everyone would play. The game would begin somewhere between the two parishes or regions and the sides would attempt to get the ball into a goal in their opponents parish. The goal could have been a door, or a wall... or a whole building. There were few rules. The ball could be kicked, or picked up and thrown. There were _no_ fouls or penalties!

This practice appears to have died out with the advent of organised sport in the late 18th, early 19th Centuries. Except in one village in England at least. At least it is the most famous case. As Ashley points out, it is the annual Shrove Tuesday (or is it Ash Wednesday?) football match in the Derbyshire village of Ashbourne. The whole town plays. Windows are boarded up. The goals are miles apart and miles wide! There are no rules other than to score, you must get the ball into the "goal". I don't believe many goals are ever scored, owing to the vast number of players, the terrain and the distances. It is still played to this day.

http://www.rsssf.com/rssbest/whysoccer.html