The Joomla Media Field input is hardcoded to be readonly, ie: you cannot edit the value of the input field, you can only change it be selectin a new image in the modal, or by removing it with the delete button. There is no option to change this.

The JCE Media Field removes this readonly property, allowing the input to be editable.

If the custom field mediajce has permission set to disabled for 'edit custom field value', the field is still writable.

I'm not sure what permission setting you are referring to here.

Edit the configuration from the mediajce custom field, change to the tab 'Permissions', select the group 'Manager' an set 'Edit Custom Field Value' to denied.

This doesn't appear to do anything for the Joomla Media field either.

I would have expected the field to be displayed as readonly to prevent tampering, but if that was intended, everything is fine.

The Joomla Media Field is hard-coded as readonly, and altering the Edit Custom Field Value parameter to Allow does not change this. I think this is a bug or omission in Joomla, although perhaps this permissions setting is used elsewhere.