Including customer details in Stripe emails
Probably best to use Craft Commerce emails, then you can put whatever information you want into the emails to your accounts departments. Like: {{ order.billingAddress.fullName }}

If the billing address information is not being sent to Stripe, you might need to modify your checkout.js or stripe.js payment screen implementation to ensure it sends the billing address information through along with the card info: https://stripe.com/docs/stripe.js#stripe-create-token

Unlike other gateways, Stripe is building the card request within javascript to make the token, when our server sends the payment request, it only uses that token - no billing address is sent again.