I have very recently been billed with another month of indesign without getting a single email notice.
I did not want to extend my plan, but the plan renewed automatically.
I quickly cancelled the plan, and asked for a refund through customer support, but it seems that they do not want to give me the refund.
Shouldn't I get a full refund if I cancel the plan within 14 days?
You have to ask Adobe. No one here is responsible for this contract.
Presumably you've passed the first month and want to cancel the second, did you sign up for the Monthly plan or the Annual plan, paid monthly? If the latter, you are committed to a one year contract. If the former you have to pay for a whole month. As Willi states, you need to sort it out with Adobe.
As you mentioned that you have canceled the subscription and have refund related query, I would suggest you to kindly Contact Customer Care for more information.
Feel free to follow up for any other questions.