After successful authentication with your IDP, if you are redirected back to the WordPress site without being logged in and if you do not see an error message then this could happen if the wordpress_logged_in cookie is not being properly set after authentication.
The miniOrange SAML SSO plugin uses WordPress inbuilt wp_set_auth_cookie() function to create the user session at WordPress.
Even though this is WordPress’s default cookie implementation, some hosting providers may block this cookie to be explicitly set and in such cases you will face this issue.
To fix this issue, please contact your hosting provider and ask them to whitelist the wordpress_logged_in cookie so that it can be set by the plugin and the user session can be created after authentication.