...and why is Ginger Ale called ale, Ginger Beer called beer, etc.

The answer is that modern soft drinks have two roots (pun intended).

Firstly many modern soft drinks come from the tradition of bottling mineral waters for tonic (health) drinks. In the nineteenth century folks started to bottle mineral spring water to be sold in cities. Then people added flavors in the form of syrups to the mineral waters. This is what you are drinking if you add Torani or Monin syrups to sparkling mineral water. Almost all modern soft drinks are syrup in carbonated water.

The other origin for soft drinks was in home brewed beverages. Farmers and other country folk made their own wine or beer. There were three basic kinds of beer made, but there was no clear dividing line between them. Small beer had very little alcohol, table beer had a bit more, then there was the stuff that would pack a wallop. The difference was mainly the time for fermentation.

Root beers, ginger ales, etc. were brewed at home with yeast (and still can be). The main difference between root beer and regular beer lies only in how much time they are allowed to ferment and when they are bottled.

Regular beer (ale for example) is made from grains and some source of sugar on which yeast cells feed (as well as nice nutrients to make the yeast happier). The stuff is allowed to ferment for a couple weeks, often with a two step process. Most of the sugar is consumed by the yeast and turned into alcohol. A little sugar can be added before bottling; this is called priming the beer. The beer is then bottled. English ale is served lukewarm while lagers are usually chilled. Why? Mainly because ale yeast works best at room tempertures while lager yeasts like the cold. In other words, they keep working in the bottle and create a certain amount of carbonation from the primed sugar.

Root beer (ginger ale, etc.) is brewed (heated) to extract natural flavors and oils from roots and to dissolve the sugars (cane, honey, molasses, etc.). Then water is added to the brew to cool it down room temperature, and yeast is added.
The brew does not sit around for two weeks like regular beer. It is immediately bottled.
The yeast multiplies fast for a few days and produces lots of carbon dioxide in the process. This carbonates the beverage which is all you want (not much alcohol). Then you put the bottles in the refrigerator. (In the old days they'd put the stuff in a cold place like a root cellar. It often wasn't cold enough and the bottles would explode!) The cold temperature stops the yeast from working. They go dormant. Home made root beer probably should not be drinken from the bottle unless you like the taste of yeast... some will almost surely be in the bottom of the bottle.
Real brewed root beer is about 0.25% alcohol (or less). This is not much, and fruit juices like orange juice have about as much.

Parsa

Quote:

...and why is Ginger Ale called ale

There was an episode of 'I Love Lucy' in which someone convinces Lucy that "Gin" is just an abbreviation for "Ginger Ale". She proceeds to get plastered.
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"It is better to have drunk and tossed
than never to have drunk at all."