I've looked through the MDN resources here as well as here, as well as the WHATWG Fetch specification, and for all that, I can't figure out if the Fetch API is part of ECMAScript 5, 6, 7 or otherwise.
All I can tell is that it isn't implemented consistently across browsers, and in some cases is not supported at all.
Yet, the specification definitely defines Fetch as JavaScript:
The Fetch Standard also defines the fetch() JavaScript API
Source: link
Is the Fetch API simply a proposal that is not on the books yet for ES7/ES8, or is it actually part of ES6/ES7/ES8 and my googling skills have failed me?
No. It's part of the Web platform API defined by the standards bodies WHATWG and W3C.
The various objects that implement the Fetch API are "host objects". i.e. objects exposed to userland JavaScript that are provided by the application hosting the runtime (usually a browser).