Destiny is like a single player game that takes place in a multiplayer world. It has similar elements to MMOs, but it's hardly the same type of game. It's drop in drop out co-op that's seamless.
Basically Bungie really loved the sandbox gameplay in parts of the Halo games and they expanded that concept into a whole game.
I think "Destiny-like" games will continue to release for the foreseeable future. It's a smart blend between open world single player games and MMOs.
It is the same type of game though. Just in first person and with a gun. The first Guild Wars would play identical if it was in first person and had guns. They innovated the shooter genre for sure, but it's not its own genre. A sub genre maybe.