It is the same.
An Xbox One home console (aka primary) that is offline will always be able to play any purchased digital games. This is also true of any Xbox 360 console.
The only Xbox One consoles that cannot play purchased digital games offline are those that are not set as home.
There are games on the PS3 that require a PSN connection to launch, no matter what. And the Netflix app, at least on PS3, won't connect to anything if you're not logged in to PSN.
If Sony has changed that for PS4, kudos to them. On PS3 if PSN is down, I can't connect to any of those apps. They all demand a PSN login on launch.