I think the amount stolen here probably precludes prosecution in most jurisdictions. But I have trouble believing that if it was a larger amount he'd be able to just say: "I was mistaken, I thought their money was my money." I'm not a lawyer but that seems like far too abusable of a defense.

According to what he says happened, this couldn't have happened with a larger amount without him realizing it.

Theft in the US afaik is also defined as "intent to deprive". That's why e.g. joyriding is different than grand theft auto.