If that iPhone is associated with your Mac's iTunes library, that's why you get that message about being associated with a different iTunes library; your Windows PC has a different iTunes library. (The Restore part of my advice is not relevant... Just an example of why I needed to use iTunes for Windows with one of my iPods.) In your case, if you have a "real" Windows PC that runs Windows 7, you can run the latest iTunes there. I guess the question now is, how to get your iTunes library on the Windows PC.
Instead of off-loading songs from iPhone (which is not part of iTunes design), you should transfer your iTunes data (the iTunes folder) from Mac to Windows PC directly. That way, you get "everything," including supporting data such as playlists, ratings, play count, data added, etc... This document describes how to consolidate and backup your complete iTunes library data to an external drive.
Manage and back up your iTunes media library - Apple Support
You can then restore from the backup to the Windows PC. One additional consideration is format of the transfer drive. A Windows PC cannot use Mac's native disk format, but a Mac can read/write some Windows disk formats; choose something like Format - ExFAT and Scheme - Master Boot Record to set up transfer drive, which both Mac and Windows PC can access.
Alternately, if you know how to set up a network connection between Mac and Windows PC, you can transfer your iTunes data that way, instead of using an external drive.
If you want to off-load song files from iPhone instead, you'll need one of those third-party utilities. I have not used one, so do not have a specific recommendation. When I change computers, I transfer my complete iTunes library from old computer to new computer. Then, it's the same iTunes library on the new computer, so my Apple devices don't see it as a different iTunes library (you should not get that "associated with a different iTunes library" message).
NOTE: For songs (and other media) purchased from iTunes Store, you can download them again at no cost
Download your past purchases - Apple Support