What's your use case? You can't use cards stored in Stripe on non-Stripe sites, if that's what you're asking.

What I was imagining is stripe issuing a credit card that my system can use to perform credit card payments on any generic webshop (either manually or some web scraper) and have that linked to a payment into my stripe account by a specific customer. I am not aware of anyone offering this service.

Why on Earth would you need to be able to do that?

Sounds like an extra layer of card fraud protection might be a use case.

Automating some shopping cases might be another.