How to sync iTunes (on iPhone) with NEW Mac...?

Any given iDevice can be linked to one and only one computer. Since you were not able to reauthorize the old computer, iTunes is acting as if you're trying to link the device to more than one computer.

You'll need to de-authorize the old computer, then authorize the new one.

These instructions are a part of a bigger knowledge base article: How to move your iTunes library to a new computer - Apple Support

1. If you plan to stop using your old computer, deauthorize it by choosing Store > Deauthorize This Computer from the iTunes menu bar.
2. If you used iTunes on your old computer to back up your iOS device, back up your devices on your new computer or manually transfer the backups from your old computer. Learn more about creating and deleting iPhone, iPad, or iPod touch backups in iTunes.
3. If you used iCloud with your old computer for syncing, set up iCloud on your new computer.
4. If you will continue to use your old computer, recreate your iTunes library. This prevents an iOS device from seeing the old computer as the new one because of the old library files.

Your i-device was not designed for unique storage of your media. It is not a backup device and media transfer was planned with you maintaining a master copy of your media on a computer which is itself independently backed up against loss. To use a device with a different setup you transfer the old library from a computer or a backup directly to the new setup, not the device to the library. Media syncing is one way, computer to device, updating the device content to the content on the computer, not updating or restoring content on a computer. The exception is iTunes Store purchases which can be transferred to a computer.

Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer - https://support.apple.com/en-us/HT201267 - "This feature works only for content bought from the iTunes Store." As of iOS9 it no longer does apps which now must be re-downloaded directly from the Store.

For transferring other items from an i-device to a computer you will have to use third party commercial software. See this document by turingtest2: Recovering your iTunes library from your iPod or iOS device - https://discussions.apple.com/docs/DOC-3991 Even this method may not fully recover what you had in the library originally. For example if in order to save space when syncing you had converted music files to a lower bitrate, or photos to a lower resolution, it is those lower quality files you will recover.

If you subscribe to Apple Music, tracks which are not part of content you have purchased or uploaded cannot be transferred and have to be downloaded directly from iCloud.

It has always been very basic to always maintain a backup copy of your computer for this very reason.

Just copy everything from your backup copy of your old computer to your new one, then sync