I wish they had a plan to literally host GHES for you because then more people in the company would be forced to reckon with how terrible GHES is from an operational perspective. It is stuck ca. 15-20 years ago conceptually.
They don't host github enterprise server for you (though gitlab has something called gitlab dedicated which they host gitlab ee for you).
Perhaps this header mentioned in the article is related, maybe that's the toggle for the enterprise mode? Seems there is at least traces of "enterprise mode" on the normal github servers.
The parent comment was talking about the "GitHub Enterprise Cloud" not "GitHub Enterprise Server" which are two distinct products.
The way that they where able to escalate the RCE from a GHES environment to github.com environment is by injecting this header and enabling this enterprise feature. This supports the idea that "Github enterprise cloud is on github.com".