The controller of The Abyss controls the ability, but the current player chooses what gets destroyed.

This has the supertype world. When a world permanent enters the battlefield, any world permanents that were already on the battlefield are put into their owners’ graveyards. This is a state-based action called the “world rule.” The new world permanent stays on the battlefield. If two world permanents enter the battlefield at the same time, they’re both put into their owners’ graveyards.