Why is it a Ginger "Beer" and not a Ginger "Cider"? It seems that Ginger Beer has no barley, no hops, no wort, no malted anything; so why is it a "beer"?  Seems like it is closer to a cider.
The term "cider" is generally reserved for apples.  There are 'pear ciders' such as Woodchuck that use the name, but as Wikipedia points out, "A similar product made from pears is called perry but sometimes (incorrectly) called Pear Cider in the marketing of some producers' products".  I personally can forgive this, since pears are closely related to apples, and the result is quite tasty.

The "beer" part of ginger beer is due to the fermentation process involved in its creation.  In this way, it is more similar to "root beer", since ginger is a root, like the sassafras root used in root beer and birch roots used in birch beer.  The website todayifoundout.com has a well referenced article for Why Root Beer is Called That covering the naming of root beer.  If the ginger based version isn't fermented, but is instead ginger flavouring added to carbonated water, it would be "ginger ale"; I have no idea why, since an India Pale Ale is certainly fermented.  Even with ginger products, the differentiation is not really all that terribly strict, as noted by this Huffington Post article.  Initially, ginger beer, root beer, birch beer, etc, were frequently alcoholic, but Prohibition ruined that for the U.S.

As a bit of a side note, the alcoholic versions are making a wonderful comeback (in my personal opinion) with Sprecher's and Small Town Brewery for root beer.  I personally have not seen as much alcoholic ginger beer, but Crabbies from the U.K. is quite good.  Although I personally do not like Kuchi's ginger beer, I have several friends who absolutely love it.