My phone says my storage is full but I am on a family plan.

A-Harris wrote:

How can I utilize the 2TB of family plan rather than my phone’s storage?

Quite simply, you can’t. iCloud storage and the physical storage on the device are two different things. You can do some things to free up space on the phone, and do some things with iCloud that will save some space for you on the phone, but for the most part, you can’t offload content to iCloud to save space on the phone.

iCloud is a syncing service, not off device storage.

This may be helpful.

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

If you have upgraded iCloud storage and are seeing a message on your device that storage is full then it is possible you are not making the best use of iCloud storage and need to use optimization features. If you do have optimization activated then it is possible you are exceeding your device's needs in terms of on-device storage and you need to buy a device with more storage.

Click on the link to learn how to optimize the storage on a device --> "How to check the storage on your iPhone, iPad, and iPod touch" - How to check the storage on your iPhone and iPad - Apple Support

- "If your device is low on storage, it automatically frees up space while installing an app, updating iOS or iPadOS, downloading music, recording videos, and more.