sepet tablonuzun adı "cart" değilse, "LEFT JOIN cart" kısmını düzenleyin.

$statement = $database_connection->prepare(
    "SELECT u.*,
        IF(c.user_id,
            CONCAT('[', GROUP_CONCAT(JSON_OBJECT('cart_name', c.cart_name, 'quantity', c.quantity)), ']'),
            JSON_ARRAY()
        ) as cart_items
    FROM user AS u
    LEFT JOIN cart AS c ON c.user_id = u.user_id
    GROUP BY u.user_id");

$statement->execute();
$userData_List = array();

while ($row = $statement->fetch(PDO::FETCH_ASSOC)) {
    $row['cart_items'] = json_decode($row['cart_items']);
    $userData_List['items'][] = $row;
}

header('Content-Type: application/json');
echo json_encode($userData_List, JSON_UNESCAPED_SLASHES), "\n";