Add custom background to div in home page

Use the wp-head-callback argument to specify your own handler:

add_theme_support( 'custom-background', array(
    'wp-head-callback' => 'wpse_189361_custom_background_cb',
    'default-color'    => '000000',
    'default-image'    => '%1$s/images/background.jpg',
));

function wpse_189361_custom_background_cb() {
    ob_start();

    _custom_background_cb(); // Default handler

    $style = ob_get_clean();
    $style = str_replace( 'body.custom-background', '#featured-home-image', $style );

    echo $style;
}