I think treatment in a private hospital wouldn't be covered by Medicare anyway. One would assume at least some treatment before or after the admission, and I think it's possible she elected to pay privately; the doctors would probably want to see her Medicare card as a backup, but provided she paid in full they would have no need to make a claim. The doctors would tend to assume that the patient would claim partial reimbursement afterwards from Medicare, but there's nothing to force the patient to do so.
However, I wonder if Marion opted for 'alternative' treatment, expected that the family would be against this and decided to avoid conflict by not telling them until after she had recovered, and this was the reason for the identity change etc. Then she didn't recover.
Yes it could have happened that another patient's treatment was processed as Marion's by mistake. I worked in a related field back then, those mistakes got made.