You are trying to add the whole Cart
object to the session.
You should just add an array, like
$cart[] = array(
'productId' => $productId,
'createdAt' => date('Y-m-d H:i:s')
);
If you need to add an object to a session, you can use __sleep and __wakeup magic functions but I think in this case it's better to just add only the product id and date to the session.