What you bought was an iCloud storage upgrade. iCloud is what holds a copy of your email, iCloud photo library, iCloud Drive, and backups. You can think of iCloud as backup of all of your data. Read up more here: iCloud storage upgrades and downgrades - Apple Support. You can downgrade your storage plan and stop paying if this is not what you are looking for. The 16 GB from your iPhone is a physical limitation of your hardware. That 16 GB is the only storage that can hold things directly on your device. If you wanted to upgrade this, you should have chosen a 64 GB or 128 GB iPhone when you purchased it.
The good news is, if what you need help storing is iTunes music you have a couple of options. Both of these will require you to be connected to the internet though. Your first option is iTunes Match. It is $24.99/year and will allow you to stream your iTunes library without taking up any of your 16 GB iPhone storage. Your second option is Apple Music which is $9.99/month but you can try it for 3 months free. Apple Music allows you to stream many millions of songs, which you don't have to own, to your device. Again you need an internet connection but if you do, you can stream the music without using any of your 16 GB of iPhone storage. https://www.apple.com/music/