Why do industrial drinks mark caffeine as a flavoring?
It could be because food ingredients are supposed to be listed in the order of their "amount":

Here's the definition from FDA:


  The ingredient list on a food label is the listing of each ingredient in descending order of predominance.
  FDA Food Labelling Guide


Periodic videos channel on YouTube showed how 6-cups of coffee leads to only 100mg of caffeine. So the amount of caffeine is insignificant by weight when compared to other ingredients (milligrams vs grams). It would likely fall below the reportable requirements if caffeine did not have the effect it does.  

For example, according to FDA "Average of 1 or more rodent hairs per 10 grams" is "OK" for ground allspice.

So it would make sense to still list it, but under the other category of aroma/flavour.

In Addition, as noted by @Sneftel in the comments, caffeine does have a very bitter flavour and adding "bitters" is obviously considered flavouring.

Interestingly, and to your point, some pain medications do contain caffeine and it's surely not listed as a flavouring or aroma on those labels.