I have a family plan but for some reasons it does not go to my accout

Have you -> Turn on purchase sharing?

When the family organizer turns on purchase sharing, they pay for family members’ purchases and must have a valid payment method on file.

1. Go to Settings  > [your name] > Family Sharing.
2. Tap Purchase Sharing, then follow the onscreen instructions.

Check here -> Share App Store, Apple TV, and Apple Books purchases with Family Sharing on iPhone

Share App Store, Apple TV, and Apple Books purchases with Family Sharing on iPhone – Apple Support (UK)

my father paid for the plan and it does not show on mine but shows on his

no it just tells me my inbox and storage is full and tells me to add my own new plan

What exactly is the problem?, it is not really clear.

and storage is full

1.

Try to free up iPhone device storage see this Apple articles below:

If you need more space for an update on your iPhone, iPad, or iPod touch

If you need more space for an update on your iPhone, iPad, or iPod touch - Apple Support

How to check the storage on your iPhone, iPad and iPod touch

How to check the storage on your iPhone, iPad and iPod touch – Apple Support (UK)

Clear Other storage on your iPhone or iPad

Clear Other storage on your iPhone or iPad - Apple Support

2.

What's the difference between device storage and iCloud storage?

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

See this Apple article: Upgrade to iCloud+ -< Upgrade to iCloud+ - Apple Support

See also: iCloud+ plans and pricing -> iCloud+ plans and pricing - Apple Support

Manage your iCloud storage

Manage your iCloud storage - Apple Support

Click here for more support and assistance.

For some reason . . .

Do you get any error message?

The problem is my storage is full

Se my previous post to free up iPhone storage.