You can pause a game nowadays?!

I'm playing a game of official chess with a time limit of 1 day. My opponent plays black. I made a move. But my opponent didn't do anything. Now there's a "pause button" in front of the one hour that he'd got left. How [removed -- MOD] is that possible?!

It is called vacation. Your opponent got placed on vacation when time ran low on the game. It is a benefit of a premium membership.

Since the game is not a tourney game, and only 1 move has been played, you should be able to abort it if you don't want to wait for your opponent to come off of vacation or run out of vacation.

Yeah, as long as it isn't a tourney game you have to have 4 moves on both sides to count as rated. Or that has been the case in the past.