No. If you learned that decades ago, then you learned something incorrect. The prior sentence contains a perfect example of commas not always appearing in pairs. It's quite common for commas to not appear in pairs, probably even more common for them not to.
A few examples of when you do see commas appearing in pairs include when such things appear mid-sentence as:
- parenthetical words, phrases, and clauses (e.g., "Sam, crying, held me.")
- vocatives (e.g., "I don't know, Ray, but I'll find out for you.")
- appositives (e.g., "That boy over there, my brother, is up next.")
- nonrestrictive clauses (e.g., "The shirt, which was his dad's, was way too big for him.")
Several examples of when you don't see commas appearing in pairs include:
- introductory modifiers (e.g., "Sadly, I won't be able to attend.")
- coordinate clauses (e.g., "I hate coming here, yet here I am again.")
- subordinate clauses (e.g., "If you build it, they will come.")
- absolute phrases (e.g., "Tom left, his jaw still smarting from Jack's punch.")
- dates (e.g., "March 10, 1970 is when he was born.")
- places (e.g., "Albany, New York is where the conference is being held.")
What appears between paired commas is most often nonessential information or not critical to the operation of the sentence, meaning you could pluck it out of the sentence and have the sentence still be completely grammatical and operational, just with a bit less information, information that is likely extraneous or trivial.
So, in short, there are situations in which commas are used in pairs, but it is not now nor has ever been a "general rule," as you put it, that "commas always appear in pairs, never by themselves."