What bank is used by Apple Card?

It is still a Goldman Sachs card. However the Apple Card does not support connections to budgeting or finance apps such as Quicken like other credit cards do. The only option they support is to download monthly statements and then manually import them into the finance or budgeting app. See: Download or export your Apple Card statements and transactions - Apple Support

Things have changed a bit since my post above. See the following from https://www.macrumors.com/2024/03/05/apple-card-ios-17-4-budget-apps/

With Mint accounts set to shut down on March 23, Apple has expanded ‌Apple Card‌ support to other third-party apps just in time for people to adopt a new service. So far, budgeting apps Monarch, YNAB, and Copilot have added ‌Apple Card‌ support.

Interesting you say this, it worked on Mint for the longest time. I connected just fine between Mint and my Apple card. (and I did not do this manually)

Now that I am using a new budget app, you are correct, it does not show up.