> Edit: Just noting that `brew install github` doesn't work, it's actually `brew install github/gh/gh` — that's a new format for brew packages I haven't seen before.

It's not new, but what you've perhaps seen is more like `brew install OJFord/formulae/gh` - 'most' (I haven't surveyed, but in my experience) people call the repo `homebrew-formulae`, `homebrew-` gets stripped, leaving:

> `<GH user>/<repo name less 'homebrew-'>/<formula name>`