Kohl, Konstantin Sorry for the delay in response and appreciate you sharing the detailed troubleshooting steps you have done so far. Based on your description, you are able to view other resources in Resources (Remote) -> Your Subscription menu, however Function app is not displayed as described here.
Azure Functions extension issues are being addressed in GitHub Repo: https://github.com/microsoft/vscode-azurefunctions/issues and please file an issue with details such as VS code version, Azure Function extension version, Core tools etc. along with screenshots (portal, VS code) if possible.
Our product team can review and assist you further. I will also help you in engaging them there.