The documentation that you've referred to is sort of a "quick and easy" continuous deployment option for Functions that gets you up and running quickly. It's not ideal for a production environment though, as you've discovered there is no way to parameterize your app settings. In order to build a proper build/release pipeline you'll need to move away from the direct Git integration and use something like VSTS which you can configure properly against multiple environments including the ability to set AppSettings and ConnectionStrings properly. Based on the approach you've taken currently, your only option is to go into the Azure Portal and manually configure AppSetting and ConnectionStrings in your function app directly.
This blog post covers a lot on the topic and shows one approach for accomplishing what you're after:
https://blogs.msdn.microsoft.com/visualstudioalmrangers/2017/10/04/azure-function-ci-cd-devops-pipeline/?utm_source=vs_developer_news&utm_medium=referral
Configuring a complete CI/CD pipeline is a bit involved (as you'll be able to see from the link) so there's no easy answer that can fit right into a StackOverflow post. I highly recommend that you read through that and do some more research on Google around the VSTS and Functions integrations.