The Plantin App website also has subscriptions on their website that Apple does not manage. You would need to cancel that subscription and request a refund with them directly for any website subscription. Only subscriptions made with an in-app purchase are managed by Apple. The way to view and cancel subscriptions managed by Apple is through the standard method here:
View or cancel app subscriptions on iPhone - Apple Support
Apple must allow links in an app to direct you to a third party website for that company to manage the subscription themselves. Previously Apple was sued for not allowing it and lost, so Apple has no choice, but to allow a company to manage their own subscription where Apple has no involvement. The fact that you said you used Apple Pay confirms that it was not an in-app purchase managed by Apple, since in-app purchases do not use Apple Pay and are billed to the credit card on file for your Apple Account.
You can certainly use Apple Pay to start a subscription with them or pay for any service to another company, but that does not mean Apple is involved in that transaction. All Apple Pay does is securely sends the unique token that was provided by the bank of the card used to the merchant. This is after you have approved the charge by double clicking the side button, followed by verification with FaceID/TouchID or Passcode. Those charges by the merchant are going directly to your card using the token you provided, and Apple Pay is no longer involved in the transactions.
The only difference in using Apple Pay and entering your card numbers directly is that with Apple Pay, the merchant does not have your actual account numbers that can be compromised in a breach and they only have the unique token to facilitate the payments with that merchant and your bank. Using Apple Pay is much safer since your credit card numbers are not exposed, but you still must deal with the merchant in the same way you would make any other credit card purchase. If the merchant issued you a refund, it would go directly to your card and does not go to Apple Pay. In fact you would still get a refund on your card, whether you still have that card in your Wallet or not.
