No, the Stripe extension does not use the Stripe Billing product offered by Stripe.
Instead, it’s designed to handle subscriptions and renewal payments by saving and using secure tokens. This allows extensions like WooCommerce Subscriptions to provide flexible recurring payment features on WooCommerce sites.
For this reason, you will not see subscriptions paid for via the Stripe extension under the Billing menu in your Stripe dashboard. Payments that are a part of a subscription will show in the Payments part of the Stripe dashboard, just like standalone orders.