Is AppStream like a server for my desktop software products?

Customers use AppStream 2.0 to stream desktop applications to their end users. Each user can access the app in a web browser or using the Windows client.

You would need somewhere to store the Quickbooks data. I believe Quickbooks has a server you run.

Here is a workshop that takes you through creating your first AppStream 2.0 environment with Chrome and Notepad++ https://appstream2.workshop.aws/

This may end up being less expensive that the cloud version. You should also consider the time investment for maintaining and managing this environment. You will also need to consider availability, backups, security and other things you get with the cloud version.

TL;DR - EC2 for QuickBooks Server, AS2 for Desktop

As @Jeremy_S said, users access applications in AS2, but QuickBooks runs a database on a "server" and that can get confusing. The server instance is decoupled from the client version of QuickBooks so as to maintain data consistency across multiple user concurrent sessions.