FOSSBilling doesn't actually save any kind of payment info.
The PayPal subscriptions work because PayPal is configured to automatically make a call to the FOSSBilling instance with specific data whenever a subscription is charged.
Stripe in FOSSBilling doe not have support for subscriptions. This is functionality we'd like to implement at some point, however due to the way Stripe handles these things we first need to upgrade FOSSBilling's own implementation of payment gateways.