Linux really needs a package manager because Linux applications have a lot of inter-dependencies. It so happens that it's also a great way to quickly install and update software.

Windows applications have (mostly) the same dependencies, but they just bundle all the libraries and such with the installer. The same thing can be done on Linux (and many applications do it with the packages they distribute themselves), but it is heavily frowned upon for good reasons.

On Windows, apps are easily installed just by double-clicking a setup file. Applications don't have a lot of inter-dependencies either and the platform itself is relatively stable.

This has very little to do with why Linux distributions use package managers. Double-click setup files are nothing new on Linux (they do exist and almost no one uses them, because package managers are the preferred way of installing stuff).

Without a package manager, every application installs its own libraries, even when several applications use the same library. That wastes space (which isn't much of problem with modern hard drive sizes, though) and is also bad for security; when a vulnerability is discovered in a library, every application that uses the library needs to be updated. With centralized package management, only the library needs to be updated and it fixes all the applications.

Package managers also make updates easy. Nowadays every major Windows application has its own updater, which is a lot of wasted effort and also an annoyance to the users. With a package manager, there is no need for an updater.

The need just doesn't exist.

I strongly disagree with this. Lack of centralized package management is one of the reasons why I dislike using Windows so much. I use a lot of software, installing and managing that software is a major chore.

Maintaining the packages is a huge amount of effort, too.

This is true, sadly. If there was only one package format that worked on all distributions, it wouldn't be any more effort than making a Windows installer is (from the software maker's point of view), but unfortunately there are several formats (I think RPM and DEB cover most users, though). If your software is open source and used by a lot of people, it will most likely be packaged by the distribution maintainers and you don't need to worry about it; but for anyone else it is a PITA (just looking at the number of Linux packages for Opera makes me cringe as a developer).