Suppose I sacrifice some Aspects of Power to Counter a card my opponent plays. All the Aspects of Power I sacrificed belong to my opponent.
Am I now susceptible to obliterate because I had cards destroyed (my sacrificed Aspects of Power)?
Yes. The trigger for Obliterate is you as a player having a card destroyed (via any means), and at the time the Aspects were destroyed, you were the controlling player.
Would he be because the destroyed Aspects of Power were his? (Alternatively, would he be because his countered card was put into his Destroyed Stack?)
Yes to the second part - your opponent would be susceptible to Obliterate because the card they were trying to play got Destroyed (via Sacrifice). Good question!