You’re right, there’s a Department of Transportation rule that says your airline must refund your ticket if you cancel a reservation made within 24 hours. The refund must be made within seven days of the cancellation.
So what went wrong? Even though you called to cancel your companion’s ticket, the agent with whom you spoke apparently never processed it. If that person had done so, Air Canada would have given you a cancellation number, which would have at least ensured your credit card dispute would go your way.
If you’d canceled your flight online, then Air Canada would have sent you a written confirmation with a cancellation number. If you didn’t have anything in writing, then you could have complained directly to the Department of Transportation. You can file a consumer complaint at its website, https://www.dot.gov/airconsumer/file-consumer-complaint.
Securing a cancellation number is helpful as some form of written verification, but it’s not the only way to verify your cancellation. Air Canada, like other airlines, records its customer service calls for “quality assurance” purposes, and can review the transcript. If you can’t reach anyone at Air Canada, try one of its executives.
I contacted Air Canada on your behalf. It decided to refund your ticket “on an exceptional basis” and promised you a full refund.