How do I backup downloaded non purchased music from my iPhone 6 to my new iPhone 13?

You don't. In order to get it on your iPhone 6 in the first place, you had to sync it to the phone from the iTunes library on your computer. That's how you get it to your new phone. You sync the music from iTunes on your computer to the new phone.

You can not transfer it directly from one phone to another.

I have the same problem, but when I connect the new iPhone 13 to the really old MacBook Pro that has my iTunes account on it, I get an error message that the phone is not authorized and cannot connect.

All of your music should be in your iTunes library on your computer. Sync your music to your new phone.