Microsoft launched the Azure cloud service back in 2010 as a way to allow developers to run added game tasks and host servers.Titanfall, for example, uses the Xbox LIVE Cloud to run dedicated servers across PC and Xbox platforms. Forza 5 uses it for Drivatars, custom AI that behaves and drives like you when you're offline.
It's a service that hasn't been entirely limited to Xbox One games, but other developers have yet to fully adopt the service. Third party developers (like Respawn) have utilized the service, but we haven't seen Sony utilize it with the PS4 (and likely won't).
Microsoft's head of Xbox, Phil Spencer, took to Twitter to restate that the cloud service was not exclusive to the Xbox One.
"Azure services are available to any game. Even other platform's exclusives. We wouldn't keep someone from using."
His response was in regards to someone questioning him on whether Titanfall 2 would be exclusive to the Xbox One, as it would be a shame to see the game lose it's cloud computing and servers. Titanfall 2 has been revealed for the PS4, PC and Xbox One sometime in late 2016 or 2017.