Why am I getting messages that my “storage is full”, and that I need to free up space when I still have over half of my storage remaining on my phone?

When you get a popup message on your iPhone or iPad that the storage is full.

Look closely storage of iPhone or iCloud

iPhone Storage (Setting > General > iPhone Storage)

1. Total storage remains same through out life span of the iPhone
2. Storage can not be enhanced
3. You may take back up on Mac or PC using iTunes without iCloud

iCloud Storage (Settings > [Tap your name] > iCloud)

1. By default every user gets 5GB of free storage space
2. You can buy more storage based on your requirement 50, 200GB, 2TB with monthly subscription
3. Based on your storage capacity you can back up iPhone or iPad
4. The space can be shared between your family members for storage plan beyond 200GB

About storage on your device and in iCloud - Apple Support

https://support.apple.com/en-in/HT206504