Short answers, pulled from the story.
The Fox Engine is a discontinued proprietary game engine created by Konami, developed under Hideo Kojima after the completion of Metal Gear Solid 4 in 2008. Its development goal was to build the best engine in the world and enable Kojima Productions to develop multiplatform games with a significantly shortened development time. The engine is named after FOX, a fictional military unit from the Metal Gear series.
Pro Evolution Soccer 2014 was the first commercially released title to use the Fox Engine, launching in 2013 on Microsoft Windows, PlayStation 3, and Xbox 360. While the engine was developed with Metal Gear games in mind, the soccer title reached market before Metal Gear Solid V.
Konami discontinued the Fox Engine in favor of Unreal Engine after seven years of use on the Pro Evolution Soccer series. The decision was driven by PES Productions focusing its development efforts on the PlayStation 5 and Xbox Series X consoles.
In March 2013, Kojima staged an appearance by a fictional director named Joakim Mogren, head of a non-existent studio called Moby Dick Studio, who showed screenshots of a game called The Phantom Pain on GameTrailers TV. Screenshots contained the Fox Engine logo, which host Geoff Keighley pointed out on air. Kojima later explained that presenting the project as unrelated to Metal Gear allowed him to observe public response to the Fox Engine's capabilities without the franchise brand influencing reactions.
The Fox Engine was first publicly demonstrated at Konami's conference at E3 2011. The demo showed a young man, a horse, and a dog in a jungle environment and was not tied to any announced game; the assets were later confirmed to have been built for Metal Gear Solid V: The Phantom Pain.
The Fox Engine supported Microsoft Windows, PlayStation 3, PlayStation 4, Xbox 360, and Xbox One. Hideo Kojima confirmed in a June 2012 interview that the engine was already running on PlayStation 3, Xbox 360, and current PCs at that time.