In the UK, appreciate that the assumption here is that we are talking about the USA - Experian offer similar services here, it is possible to tell a credit card company to block reoccuring payments from an individual merchant. Its not used often, many banks are bad at training their staff on how to do it etc but it is indeed possible. It isnt possible to block a particular contract though (so if you paid your Auto and Home insurance via AIG you could block AIG or not, you cannot block it for the Auto cover but allow the Home to continue). I am surprised you cannot do similar in the USA.

It is however the same here in that cancelling payment is not the same as cancelling a contract and not only are you potentially still liable to pay it but potentially adding late payment fees etc to the bill. However subscription based services are often difficult to cancel and intentionally a pain which is exactly why card issuers had to introduce the mechanism of cancelling reoccuring payment authorisation in the first place.