We do not know why the soft drink ginger ale was called “ale,” since ale is an alcoholic beverage that comes from the distant past and has already gone into an unforeseeable future (see Star Trek and the popularity of “Romulan ale” there).
The word “ale” (Old English ealu), which designates an intoxicating liquor made by malt fermentation, has come into English from its ancestor language, Common Germanic, where the word may have been *alúþ-. Other Germanic languages like Middle Dutch, Old Saxon, Old Norse had words connected to “ale.”
The ultimate origin of the word is still disputed. It seems, however, that the Germanic word derived from the Proto Indo-European root *olú-t- (from an earlier base *h₂elut-), which originally meant “golden or reddish color.” This word *olú-t- then came to refer specifically to “ale” because its color is golden, originating both the Germanic word *alú-þ- and the Ossetic word æluton (“a kind of beer”). The Proto-Indo-European word olú-t- may have also been inherited by Slavonic languages (Old Bulgarian olu “cider,” Slovenian ol “beer”) and Baltic languages (Lithuanian alus, Latvian alus “beer,” Old Prussian alu “mead”). It may have been also borrowed by non-Indo-European languages like Finnic languages (non-Indo-European), giving Finnish olut and Estonian õlu, and Georgian aludi (“a kind of beer”).
All these are fascinating tidbits, but how they relate to the Armenian language?
You need to turn to the current designation for distilled alcoholic beverages like vodka or raki: օղի (oghi). Incidentally, the work բարկօղի (pargoghi), literally meaning “bitter oghi,” is a modern Armenian name for another distilled beverage, cognac or brandy.
The word oghi word already designated a “strong drink” in the Bible (Luke 1:15, King James Version), but when we go to the fifth century Armenian translation, we have to remember that the Armenian letter o did not exist at the time but was a late medieval replacement for the diphthong աւ (aw), and the sound gh denoted a stronger l. Thus, the Classical Armenian form was աւղի (awghi)/awli, which may have come from *alu (< *olú-t) with the change of alu into awl(i) as the result of a phonetic process technically known as metathesis. This form awghi showed in the first centuries of Armenian writing as եաւղի (yawghi) and ուղի (ughi) too.
Over the centuries, then, “ale” and oghi diverged in their meaning, with one used for colorful beverages, and the other, for clear ones. But they met halfway, since their alcoholic origin has kept them popular among drinkers.