Basically the author wants to have a Chrome OS installation that can run android and non-root unix/linux software. All this in a regular installation without requiring the user to enable developer mode.
Regarding your questions, yes it is running linux as Chrome OS is based on Gentoo. But it is not running any traditional linux distro. In order to provide a linux/unix environment, the author installs termux. But this app is for command line programs. The quake screenshot has nothing to do with termux, it was installed from the Play store. Therefore I'd say it's not possible to run Steam on it.
Regarding your questions, yes it is running linux as Chrome OS is based on Gentoo. But it is not running any traditional linux distro. In order to provide a linux/unix environment, the author installs termux. But this app is for command line programs. The quake screenshot has nothing to do with termux, it was installed from the Play store. Therefore I'd say it's not possible to run Steam on it.