How to transfer my apple music songs from my iPhone 6 onto my new Macbook Pro

Hello Erik,
Welcome to the Apple Support Communities and congratulations on your new MacBook Pro. I understand that you are trying to get some music that is on your iPhone on to your computer. That is something that I can help you out with.
The first thing I want to point out is that when you try to transfer items that have been downloaded from the App or iTunes Store's, that content is the only content that moves over. This would be the reason why only the purchased content from iTunes was downloaded. Take a look at the first article for more details about transferring purchases.
Redownload or transfer your iTunes Store purchases from an iPhone, iPad or iPod to a computer
https://support.apple.com/en-us/HT201267
If you are using Apple Music and have iCloud Music Library enabled on your iPhone, then I would recommend checking is the iCloud Music Library settings in iTunes. With that enabled, you will be able to keep your library in sync without plugging in your iPhone to your computer all the time. Check out the second article for the steps to make sure iCloud Music Library is enabled.
Check your Apple Music library settings
https://support.apple.com/en-us/HT204926
The final suggestion would be to copy the music from your old computer to your new Mac. If you still have access to it, you can use the steps outlined in the third article to copy all the content over with ease with an external hard drive.
How to move your iTunes library to a new computer
https://support.apple.com/en-us/HT204318
Have a great day

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.

iTunes Store: Transferring purchases from your iOS device or iPod to a computer - http://support.apple.com/kb/HT1848 - only purchases from iTunes 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 you had converted music files to a lower bitrate, or photos to a lower resolution, in order to save space it is those lower quality files you will recover.

The best solution, as indicated in another post, is to transfer the entire library intact from the old computer to the new one.

Hey Erik,
Glad to know that I could help and you got your music.
Take care