By the strictest old standard beer is made from only three ingredients: water, germinated barley (malt), and hops. Yeast was not understood at the time and like with sourdough bread, naturally occurring yeast spores found their way into the beer brewing process but not deliberately as added ingredient. Eventually, Very likely, some batches of beer were much better than others and the brewers weren't dumb and knew something was missing. They began adding something akin to a yeast starter to the brewing process, and by the early 20th century the use of directly adding yeast as the fourth ingredient became legal and common practice.
If sake were made from barley instead of rice and with hops it would be considered beer. It's not wine either but its own thing.
The alcohol in Hard Mountian Deww is very likely made from sugar, malt, and fruit. Not beer.