Cookies in Multisite network where sites have their own domain name?

First clear your browser’s cache (including cookies) and your server’s cache (cache plugins, etc.) Then set the following in your wp-config.php file:

define('ADMIN_COOKIE_PATH', "");
define('COOKIE_DOMAIN', '');
define('COOKIEPATH', '');
define('SITECOOKIEPATH', '');

Also, you may checkout the answer from HERE:

define('WP_ALLOW_MULTISITE', true);
define('MULTISITE', true);
define('SUBDOMAIN_INSTALL', false);
define('DOMAIN_CURRENT_SITE', 'your-domain.example');
define('PATH_CURRENT_SITE', "");
define('SITE_ID_CURRENT_SITE', 1);
define('BLOG_ID_CURRENT_SITE', 1);
define('SUNRISE', 'on');

If it still fails, then read this answer or contact your server support, there may be a configuration issue in the server.