It's not a sharepoint issue, it's an ASP.Net issue. ASP.Net only allows one form. Yes, this is bizarre to folks who are new to it, but that's the way it works and has worked for over a decade now. So, perhaps:
- Redo your logic to work with one form.
- Put your custom functionality outside of the ASP.Net form
Or, as another possibility, I don't know if it's required to use an ASP.Net form for SharePoint hosted apps? Anyone else know that? (It isn't required for a provider hosted app.)