Besides, if one were to be guided by their kernel of choice, then the PS4 would have a very crappy GPU support :D

Could you simply develop in OpenGL on Linux and compile for Steam and PS4?

As above, there is no OpenGL implementation for the PS4...

Direct3D 9 gets you WinXP+.

Direct3D 11 gets you WinVista+.

Direct3D 11.1 gets you Win8+.

OpenGL2 + GL3 + GL4 gets you Linux/Mac/Windows.

GNM gets you PS4.

GCM and/or PSGL gets you PS3.

GXM gets you PSVita.

"DirectX 11.x" gets you Xbone.

"DirectX 9.x" gets you Xbox360.

GLES 1 + GLES2 gets you Android/iOS.

WebGL gets you HTML5.

GX gets you Wii.

GX2 gets you WiiU.

If you want a cross platform graphics API, you've got to make your own wrapper around most of the above... or you can use an existing game engine that's already done this work

Could you simply develop in OpenGL on Linux and compile for Steam and PS4?

As above, there is no OpenGL implementation for the PS4...

Direct3D 9 gets you WinXP+.
Direct3D 11 gets you WinVista+.
Direct3D 11.1 gets you Win8+.
OpenGL 2.x/3.x/4.x gets you Linux/Mac/Windows.
GNM gets you PS4.
GCM and/or PSGL gets you PS3.
GXM gets you PSVita.
DirectX 11.x gets you Xbone.
DirectX 9.x gets you Xbox360.
GLES 1.x/2.x gets you Android/iOS.
WebGL gets you HTML5.
GX gets you Wii.
GX2 gets you WiiU.

If you want a cross platform graphics API, you've got to make your own wrapper around most of the above... or you can use an existing game engine that's already done this work

And Mantle gets you AMD.

And Mantle gets you AMD.

Yep. On PC.

FWIW though, if you've got a PS4 game, porting from PS4->Mantle will be much easier than porting from PS4->Dx11 or PS4->GL (AMD did make the PS4 API, after all).

Also, PS4/Mantle allow you to do things that simply cannot be done via Dx11/GL!

It would therefore be impossible to port some features from console to Dx11/GL, without re-implementing them using a different algorithm. Mantle on the other hand allows for a straightforward console->PC port (for AMD users only...)

Usually when developing a console game, you actually do most of the work on PC. You use an engine that runs on the PC and on the console, so that most people on the team can work with just a PC and only occasionally test/profile on the console.
This presents a problem when you're using features on your console that cannot be implemented on your PC build.
Mantle solves that issue by bringing console-like capabilities to the PC.
It wouldn't surprise me if many console-only developers start supporting Mantle internally just for their own development tools