But you're not forced to do it in order to proceed in the "story" of the game, as most people understand it. You have to find your own motivation to do it (which is a huge positive, I think).
How is Dagna's quest any different from a fetch quest? We are given a task to go somewhere and do a thing and then report back?
Right, it's not necessary to the main plot, and as such is side-content. However, it is 'story-driven', in that a scripted story is expressed, and the primary point of the activity is to progress through that scripted story--that's as opposed to, say, astrarium-solving, in which the primary point of the activity is to solve astrariums, ultimately unlocking some loot. This is all semantics, of course, but the main plot isn't the only 'story' in the game, and the point of even making the definition between story-driven and non-story-driven content in this case is to establish the difference between quests like the companion stories, and activities like shard hunting.
What's the difference? How is this different from collecting the Grey Warden artifacts for Blackwall?
Because it's far more focused - I converse with Dagna, I converse with the archmage, then I converse with Dagna again - and less about simply filling up a bar/ticking items off a list? If Dagna asked me to go collect twenty bits of nug poop so she could disguise herself as a barrel of manure to escape Orzammar, there'd still be a 'story' to the quest, but the ratio of story to tedious gameplay - which, to me, collectable hunting tends to be - would be far greater. Technically speaking, there's even a story to all the shard hunting in DA: I, but that story is incredibly thin, proportionate to the amount of time you have to spend to complete the activity, as contrasted with most - though not all - pre-Inquisition BioWare side content. There's clearly a divide between the two sorts of content, or so many people wouldn't have noticed - and commented upon, either in approval or disapproval - DA: I's preponderance of 'filler' following its release.