Daily.co's video calling API uses Stripe for credit card processing and billing, with a custom implementation that integrates Stripe's recurring billing features into their own system. The implementation involves mapping between Daily.co's user and team data models to Stripe's customer and subscription records, as well as creating a data model that translates usage data into Stripe's metered billing format. This requires handling issues such as payment processing, invoice creation, and plan changes, with the goal of providing a clean and maintainable codebase. The system uses webhooks to receive notifications from Stripe about events such as invoice creation and payment success, allowing for automated updates to the underlying data models.