Backing up all the music files is always a good idea. I would use Windows Explorer, not Windows Media Player, and Copy them from the Folders where they are to wherever you want them backed up.
In Windows Media Player, Right click on a song that you want to back up, and click on Properties:
That will Open Windows Explorer, showing the Folder containing the song.
If you want to Copy every song in the Folder, you can hold the Ctrl key and touch the A key on the keyboard, and it will select All the files in the Folder. Hold Ctrl and touch C, and it will Copy all the files in the Folder.
Select (or create) another Folder, or another Drive (Flash Drive or External Hard drive, whatever works). Navigate to that Folder in Windows Explorer and then just hold Ctrl and touch P. All of the files will be pasted in to that new Folder. The originals stay where they were, er, are.
I am Old School. I much prefer to manipulate, rename, and change Properties of Files using Windows Explorer instead of Windows Media Player. It does not confuse the WMP Libraries so much.
Notice above that the Filenames have numbers and the Titles do not. That is because the old, tired, CD player in the car sorts the files by Filename. Unless I want to listen to them in alphabetical order, the Filenames need to be numbered.
The quickest way to re-name a File is to double click on it (Pushing F2, Function key F2, does the same thing.) If a file is inadvertently double-clicked on, it will look like this:
Windows Explorer thinks the User wants to rename it. If you don't, just hit Esc at top left on the keyboard and all will be forgiven.
This renaming method is what I would have done to remove the numbers from the Filenames.
Remember back when that was all that you wanted to do?