Earlier quoted context omitted.
The standard library is usually delivered as part of the operating system on *nix-like platforms.
Making statements as inaccurate as possible without being wrong is a fun game, I guess. My turn: A file explorer is part of the OS because it is installed by default on Windows. A good OS provides a graphical file browser.
For example, on Solaris and various *BSDs, the syscall interface is private or unstable and is explicitly NOT an interface.
So without the standard library, none of the applications could run.
That sure sounds like part of the operating system to me, and fits various accepted definitions such as the one Wikipedia provides.
And yes, I would consider the file explorer included with Windows part of the operating system, as most people I think would also.
The confusion here seems to come from the Linux world, where components are mix and match; where you can pair the kernel with an entirely different base.
That isn't true of many other operating systems; the base distributed set is designed to work together and provide the environment and platform for applications.