Wrapping a Sidebar within

UPDATE: now that the post is clarified, in your custom sidebar declaration change these lines to the following (this preserves existing wrappers too, which are advisable to leave in unless you know you want to delete them):

            'before_widget' => '<div class="customdiv"><div class="widget-content">',
            'after_widget' => '</div></div>',

---

I assume you’re trying to edit a system-loaded sidebar. Try this code in functions.php. The sidebar you want to edit with this is going to have an id on the front-end that you can find by using the tag inspector in Developer tools in your browser, and once you have that id you have to replace ‘specific_sidebar_id’ on the fifth line below with the id of the sidebar you wish to alter.

function customize_my_sidebar() {

    global $wp_registered_sidebars;

    $key = array_search('specific_sidebar_id', array_column($wp_registered_sidebars, 'id'));

    $wp_registered_sidebars[$key]['before_widget'] = '<div class="customdiv">';
    $wp_registered_sidebars[$key]['after_widget'] = '</div>';

}

add_action( 'wp_loaded', 'customize_my_sidebar' );

If you’re registering an entirely new sidebar, you should customize “before_widget” and “after_widget” in your definition when calling register_sidebar() as documented here: https://developer.wordpress.org/reference/functions/register_sidebar/