I want to store the created sessions in a directory above the root, except when I use any of the following:
session_save_path($_SERVER[\'DOCUMENT_ROOT\']
PHP requires session file to be owned by the user running PHP.
Then just run chown -R www-data:www-data /var/www/storage/sessions/ to own session folder and files by your PHP user (www-data:www-data are your PHP user and group separated by :).
You can use PHP method get_current_user() or run in bash php -i | grep user and find your user running PHP there.