Most people I know never called that a file-browser in the first place. It was just "my files", as if the files were things in their own right that just appeared on screen. OS vendors have played into that, with icon labels that say "My Documents" or "All my files". It's to their advantage to not let anyone know that there's a separate program that sits on top of the filesystem to show the contents of the filesystem. Most people don't care how things work as long as they work.

For that matter, I know a number of folks who don't know what a web browser is either. It's just "The Internet", thanks to Microsoft's marketing, or "the Googles", thanks to Google's. When I used to read Google user feedback, it was amazing how many people thought that anything that happened online or offline due to a Google Search was Google's fault - we'd get feedback like "The plumber came 2 hours late...I want my money back!"

The term in wide use when I came into the field seemed to be "file manager" for an application that was about files. For a file selection dialog box, I've heard "file browser" from time to time.

Technically a browser isn't even a full application. It's part of the interface of a bunch of different types of applications. A "web browser" is a file renderer, a content fetch utility, an object container that's targeted by a built-in scripting language, and a whole lot more. One of its functions - just one - is to follow hyperlinks and let you browser / surf for other content. And yet "browser" means, yes, "web browser".

This is not an argument about what the word means or should mean. It's an admission that we allowed the word to become specialized. Now it's either go with the status quo or try to take the word back, like "hacker", or maybe "Kleenex", "Aspirin", or "Rollerblade".