They actually did make fetch happen. Once upon a time it was usual in Javascript to use a thing called XMLHttpRequest which despite its name isn't actually for XML, it's just that XML was a big deal when it was created. The replacement API for making normal HTTP requests is just named fetch, and it was "new" so long ago that popular web browsers had versions like 40 rather than 150.

That movie is so old it's entirely possible that it's just named "fetch" because that's a reasonable thing to call this feature and so it's a coincidence, but I do like to think that at least some people at WHATWG were quoting Mean Girls...

I worked for Fetch Robotics (now defunct), and there were a bunch of people (especially in management) who would constantly reference the Mean Girls "stop trying to make fetch happen" line in company-wide slideshows.

A couple of times it was cute... but they took it too far in my opinion. And sadly the company was bought out, and now they too have decides to "stop trying to make Fetch happen" (yes, officially it was bought out, but not for the actual robots part).