御教示いただいたサイトの記事を参考に

OrderTrait.php

<?php

namespace Customize\Entity;

use Doctrine\ORM\Mapping as ORM;
use Eccube\Annotation\EntityExtension;

/**
* @EntityExtension("Eccube\Entity\Order")
*/
trait OrderTrait
{
/**
* @ORM\Column(type="string", nullable=true)
*/
public $department_name;
}

というファイルを作成し、
../../../bin/console eccube:generate:proxies
を実行し、
../../../bin/console cache:clear --no-warmup
をやったら
PHP Fatal error: Eccube\Entity\Order and Customize\Entity\OrderTrait define the same property ($department_name) in the composition of Eccube\Entity\Order. However, the definition differs and is considered incompatible. Class was composed in /var/www/html/eccube/app/proxy/entity/src/Eccube/Entity/Order.php on line 42
In Order.php line 42:
Compile Error: Eccube\Entity\Order and Customize\Entity\OrderTrait define the same property ($department_name) in the composition of ccube\Entity\Order. However, the definition differs and is considered incompatible. Class was composed

となってしまいました。

src/Eccube/Entity/Order.php

に
/**
* @var string|null
*
* @ORM\Column(name="department_name", type="string", length=255, nullable=true)
*/
private $department_name;
/**

の記述があるのが重複しているということでしょうか。