I don't think it runs Linux, there is tons of embedded OS's.
It could be Nuclus, RIOS OS, or most likely, some sort of proprietary OS which was shipped with specific MCU they are using.
OS writing is pretty simple, as long as you do not need a ton of device drivers and networking. There are over two dozen of them listed there: https://en.wikibooks.org/wiki/Embedded_Systems/Common_RTOS and that is not a complete list.
It’s amazing that this OS also has its own compiler and runtime capable of running Quake translated into Lisp.