"which suggests it is his own account"

They are only and always Individual Owned. That's the point of Retirement accounts.

If he thought he was putting $6,000 x 2, you need to indicate this as there would be two different years for Form 5498. Get the 5498 to see what happened.

"Are you able to contribute $12,000 (for a two year period)"

As two separately labeled events. Right Now, you can fund your 2021, if you waited, and your 2022, for instance. But you Tag Them as such.

"for a backdoor Roth?"

"Backdoor" just means, Converted immediately to Roth, to avoid earnings that would be taxable, and the contributions were not deducted, so they are Basis, and conversion of Basis is not taxable.

"Assuming the answer is no, is he allowed to contribute $6,000 each for him and his wife into his one account?"

Nope. He puts her share into Her account.

"into two separate 1099-Rs for $6,000 (for each spouse)."

You didn't issue the 1099-R. You don't get to change reality.

There still is time to do a Corrective Distribution, and move her half to her account.