I don't know how else I could've done it...
... by visiting their web page in your browser, as you did with Itch? ;-)
If you create the account with the app makes no difference. The account is useable by website. And it has to. If you pay with paypal on a shop that has this, they will redirect you to the paypal website. Once the payment is approved by you, after logging in with your credentials, the website redirects back to the shop, but tells the shop that the payment has been taken care of.
The reason this approval works with credit cards is trust. There are things like chargebacks. And for debit cards that can be an issue, if the balance is negative. I think this is why it is a thing to advertise you accept debit cards.
When entering your "card" number, did you only entered the number, or a code on the back side plus expiration date of the card?