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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Who cares about what genre it is? It's fun