Cannot Update user meta in custom field

Welcome, either use edit_user_profile_update which is passed user_id

add_action('edit_user_profile_update', 'save_extra_fields_country_institution_user_form' );
    function save_extra_fields_country_institution_user_form($user_id){
        # again do this only if you can
        if(!current_user_can('manage_options') || !current_user_can( 'edit_user' ))
           return false;
        # save my custom field
        $result = update_user_meta($user_id, 'verifier_assign_country', 'test3');
    }

OR
when I tested on ‘edit_user_profile’, the whole user object is being passed, not just the user id, so use:

function save_extra_fields_country_institution_user_form($user){
    # again do this only if you can
    if(!current_user_can('manage_options') || !current_user_can( 'edit_user' ))
       return false;
    # save my custom field
    $result = update_user_meta($user->ID, 'verifier_assign_country', 'test');
}

However https://codex.wordpress.org/Plugin_API/Action_Reference/edit_user_profile is more aimed at adding fields to screen, so I think what you want is ‘edit_user_profile_update’