Transfer Music to a new Mac and keep all Playlists

Hello WALTER-MILANO-ITALY, 

We understand that you have a friend wanting to migrate their Music library to a new Mac and referenced Back up and restore your iTunes library on your PC.

The original article you referenced is about moving the library on a Windows PC and would not necessarily apply in our situation. 

iTunes was replaced with the Music app in macOS Catalina. That means since both computers are running macOS Big Sur there is no iTunes app available on either computer. 

We can certainly provide a few articles on moving files though. 

If they are setting up their new Mac for the first time, they can use Migration Assistant to move their files and organization Move your content to a new Mac. This is going to move the files they want in their user account from one computer to another and the new Mac would be set up with the same details as their current Mac. 

Your friend can also copy their Music library folder from where it is on their originating Mac, move it to the second Mac and then point the Music app to that location. That means the Music app will read the library from those specific folders and files - Change where your music files are stored on Mac. 

It’s always a good idea to have a Time Machine backup of their data, especially since they have taken the time to organize their data just the way they want it. Back up your Mac with Time Machine

Let us know if either of these options work for them.