No matter what ever way you look at the game, Destiny is a MMO.
Is it Massive? Yes, there are tons of people in the hub and in the events.
Is it Multiplayer? Yes, that should be a no brainer.
Is it Online? Yes, it forces you to be on the internet just like a normal MMO.
So why do I see people calling it a "Shared World Shooter"?
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Have you seen how many sub genres there are in music? They call it something different because when you think of MMO you think of World of Warcraft or DC. That type of game. This is different. The Division is not called an MMO but it's practically the same?