An HSA might have been established through an employer, but it's his personal account. It isn't like a 401(k) that gets "left" with the employer. It's a banking account, and it's personal, so he needs to figure out who has his account. You can use that same HSA account as you change jobs, as long as your coverage still qualifies you for an HSA.

He's been spending from it, so he is misleading you. If he has a spending card, turn it over.