'Contact' field on the Experience Site 'Create Case Form' or 'Contact Support Form' Auto Populates with Customer Community User

This behavior is by design in Experience Cloud. When a Customer Community Plus or Customer Community user owns a case, the Contact Name field on the case must reference the contact associated with that same user. You cannot specify a different contact, even if the alternative contact is associated with the same portal account.
Exception — Portal Super User Permission:
Users with the Portal Super User permission are exempt from this restriction. The Portal Super User permission allows delegated external user administrators to create cases for contacts other than themselves within their own account.
Note: If a case created by an External User is edited by an Internal User, the contact automatically switches back to the external user.
How to enable the Portal Super User permission:

1. Navigate to Setup.
2. Go to Users > Profiles.
3. Open the profile of the Customer Community Plus User.
4. Click Edit.
5. Select the Portal Super User checkbox.
6. Click Save.

Alternatively: Navigate to Setup > Users > Profiles > Profile of the Community Plus User > App Permissions > Edit > select Portal Super User > Save.

See Also:
Grant Super User Access to Customer Users in Your Community