Issues Updating Post Meta with AJAX (Seems simple but cannot figure it out)

You did some mistakes that I have described below:

First thing you have placed jQuery Ajax code on the wrong place.
On ajax callback function variable name mashup like $post_id defined and used on update time $post_ID; PHP is case sensitive so it considers 2 different variables.
In some places, not closed } brackets.

Here is the right code which I have modified in your code.
Check this and let me know it’s work for you or not.
I have tested it works for me.

<?php /*Register the Menu*/ 
add_action( 'admin_menu', 'ews_metas_add_admin_page');
function ews_metas_add_admin_page() { 
// Adds plugin page to menu
add_menu_page ('Effective Web Solutions Metas Descriptions', 'EWS Metas', 'manage_options', 'ews_meta_options_stephen_version', 'ews_meta_page_creation_stephen', get_template_directory_uri() . '/images/admin-ico-gray.png', 110 );
} 

/*Page Html Render*/
function ews_meta_page_creation_stephen() { ?>
<div>
<?php settings_errors(); ?>
<form method="post">
    <input id="meta" class="testing" type="text" name="2316" value="<?php echo esc_html( get_post_meta( 4189, '_metadescs', true ) ); ?>" />
    <?php submit_button();?>
</form>
</div>
<script>
jQuery(document).ready( function($){ 
jQuery(".toplevel_page_ews_meta_options_stephen_version form").submit(function() {
        var post_meta = jQuery("input.testing").val();
        var post_ID = jQuery("input.testing").attr("name");
        jQuery.ajax({
            type: "POST",
            url: 'https://dev.ewsproduction.com/dev6/ewstheme/wp-admin/admin-ajax.php',
            data: {
                action: "update_meta_descriptions",
                post_id: post_ID,
                post_meta: post_meta,
            },
            success: function( data ) {
            //do something
            console.log('It worked!');
        }
    });
    return false;
    });
});
</script>
<?php }

/*Update the meta using Ajax request */
add_action( 'wp_ajax_update_meta_descriptions', 'update_meta_descriptions' );
function update_meta_descriptions() {
  global $wpdb;
  $post_id = $_POST['post_id'];
  $meta = $_POST['post_meta'];
  $details = update_post_meta( $post_id, '_metadescs', $meta );
  echo 'Meta Updated';
  die();
}?>

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