XboxOne uses DirectX and Direct3D and/or custom to hardware
PS4 uses from what I know OpenGL and/or custom
I really hope, that AMD takes and really launches something but they are just another Microsoft company not really focused on GNU/Linux at this moment only releasing fixes/patches to big issues and listening to their enterprise customers not so much desktop.
Nvidia seems to focus on OpenGL and moving more focus to Linux while AMD just looks at it and leaves small breadcrumbs hinting that Mantle is able to launch for Linux "when the market is there". In this pace, the mantle market will probably drop more and more and if not will be gone when OpenGL/Direct3D API's solve or mitigate the issue that Mantle is there to fix.
I really hope that Mantle would launch and give some initiative for vendors to implement ant give one holy API that works on Windows/Linux but given the above and their current pace...im not so optimistic that it will ever launch.