Diablo 3 is an always-online game; the reason is that you can take the items you earn in single player and use them for your multiplayer character (and vice-versa). Their security is extra tight because up until recently, said items could be traded for real money. And you don't want anyone to be able to generate things like that offline.

Although they did, there were some item duplication bugs. And even that was irrelevant, since there were quickly popping up small armies of item / gold farmers; every once in a while they'd get banned, but by then they would've earned enough money to buy another fifty copies of the game and continue.

Is Diablo III single player? When I played it at the release, connecting to a remote server was required (which is a shame, btw. I will miss the modding community that Diablo II had).

It can be single-player, but you still need a connection to do so.

Is this also true for the console port(s)?

It is multiplayer only (always connected) but you can play it alone.