I'd argue that many mmo's have story but destiny isn't an mmo either.
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Many MMO's have quests (bounties) not really a story. Massively Multiplayer Online, do the masses play, check, can you play with many other people, check, is it online, check.
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Massive the game supports a standard party size like that of call of duty or any modern fps. Take a look at wow or guild wars or Star Wars the old Republic. All these mmo's have story's that many play just for that. Destiny is not a mmo it borrows aspects and that's all. So destiny is just an online shooter.
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I would say Counter Strike is an online shooter, Destiny is way more than that. The only recent MMO that I would agree that really raised the bar on story telling would be SW:TOR. The rest, meh. Look at Sony's Planetside 2, arguably less content than Destiny, pvp only, individual classes that level up as you play and it's considered an MMO FPS Shooter, so why not Destiny?