Yes, so? That doens't mean that Stripe isn't the merchant of record. All that matters is whose name shows up on the credit card statement. If that's Stripe - as it usually is because it is their merchant account - then that's Stripe. Bring your own merchant account and it will be your name that shows up. Stripe can be IPSP and PGP depending on the circumstances and the story the OP told here strongly indicates that they are using Stripe as an IPSP.
The difference between a payment service provider and a payment gateway is substantial, in the first case you likely won't ever see the customers card details other than what is required for shipping, in the second you will have all of the data including CVV and expiry and you will have to deal with the associated overhead.
The difference between a payment service provider and a payment gateway is substantial, in the first case you likely won't ever see the customers card details other than what is required for shipping, in the second you will have all of the data including CVV and expiry and you will have to deal with the associated overhead.