> But "ginger ale" seems straightforward to me. It's an ale, flavored with ginger

Ginger ale is in fact, not an ale, it's a soft drink. It is distantly related to Ginger Beer and some variants of Ginger Beer are alcoholic like ales, but Ginger ale was conceived as a soft drink and today continues as a soft drink.

My (likely ignorant) understanding was that the non-alcoholic "ale"s and "beer"s were fermented like their traditional versions, but the process was stopped before the ethanol level became significant.

Mass-market ginger ales and root beers are not made that way today, of course.