Displaying which Role the current user is assigned to

As you suggested, here’s how you can display the roles next to the username in the admin bar:

function wpse_203917_admin_bar_menu( $wp_admin_bar ) {
    if ( ! $node = $wp_admin_bar->get_node( 'my-account' ) )
        return;

    $roles = wp_get_current_user()->roles;

    $node->title .= sprintf( ' (%s)', implode( ', ', $roles ) );

    $wp_admin_bar->add_node( $node );
}

add_action( 'admin_bar_menu', 'wpse_203917_admin_bar_menu' );