The most likely cause of this problem is that the songs haven't been downloaded or transferred into iTunes from your 5S.
I would plug in your 5S and then click sync; if it asks if you want to transfer purchases, click yes.
Now when you plug in your 6S, all of the content should be iTunes, and all you have to do is sync again. (This is all assuming that the sync settings are the same between the 5S and 6S) What I mean is, if you told iTunes to add 77 albums to the 5S, but then changed the settings to only upload 35 albums, it will only add 35 albums to the new iPhone.
When you plugged in the iPhone 6S, did you restore from an iPhone 5S backup? If you did, the devices should mirror each other as long as you have a current backup and are not using and old one as your point of restoration.