How to back up?

iTunes is a one-way street - it only copies songs FROM your computer TO your iPhone. It cannot copy songs FROM your iPhone TO your computer.

If you want to copy songs FROM your iPhone you need to get a third-party utility such as iMazing or copytrans (this is not a recommendation, however, it's just two of a number of utilities that can copy data FROM your iPhone TO your computer).

If your old computer is still available, you could migrate your iTunes library from your old computer to your new computer - that would do what you want to accomplish.

Follow this link

Sync your iPhone, iPad, or iPod using your computer

Learn how to sync music, movies, and more using your Mac or PC.

https://support.apple.com/en-us/HT201253

Have you logged into your Apple ID with the same one used on the iPhone and authorized this computer with iTunes?

Something is wrong then because your phone thinks you are using a different Apple ID library on your computer.