The word "ale" has a variety of different specific meanings (e.g. beer brewed without hops, or beer brewed using warm fermenting yeast, or beer with a higher alcohol content than lager, or sometimes just any kind of beer). But all of those definitions refer to a type of beer -- an alcoholic beverage brewed via fermented grains.
I can see why ginger beer is called that: it was originally brewed and fermented like beer, just with ginger instead of a grain, and did contain as much as 10% ABV (though these days it's not brewed anymore and is non-alcoholic). But I can't find any evidence that ginger ale ever contained alcohol, or was ever brewed like ale.
So how did it get the name ginger ale?