Why is my iPhone storage full if I have plenty room on iCloud

It is not possible to add storage to an iPhone; it will never have more than it did when you bought it. What you bought is iCloud+ storage that is used for backing up and also for syncing to other devices. Here is an explanation→What's the difference between device storage and iCloud storage? - Apple Support

iCloud+ has 2 independent functions. iCloud backups, and iCloud sync. It’s confusing because they both have the same name prefix. But they have no connection with each other. Go to Settings/[your name]/iCloud. You will see a bunch of switches (with iOS 16 also tap Show All). When you turn on a switch that data type will sync to iCloud. If you do this on multiple devices that share an Apple ID the selected data will sync to all of those devices. 

You can also turn on iCloud Backups. This will back up your phone every night if the phone is plugged in, connected to Wi-Fi and locked.

But note that anything that you sync by the first method will be excluded from the iCloud backups.

Sufficient local storage is required for all iPhone operations.

iCloud provides a place to off-load stuff, but does not increase local storage. What an iPhone ships with is what it will have.

What's the difference between device storage and iCloud storage? - Apple Support