In the US, yes. If the letter of the contract says that you are responsible for a monthly payment, how that debt is settled every month does not change the terms of the contract. A gift card just allows you to settle the debt with a mutual debt owed.
I was mostly wondering if the contract is indeed the same or if the services have a separate fixed-term contract for gift cards. But I guess it’s par for the course (ie. really customer-hostile) if they indeed force even gift card paid subscriptions to be perpetual…