This publication profile is the user for deployment. It also generates the password by the Azure platform itself. Why is there a dollar worth and so on - a mystery covered in gloom. It is known, probably only to developers of the platform. There are some guesses why this is so, but no more.
If you are going to deploy your application for example via FTP or TFS, and even just from Visual Studio, then this password and user will be asked for you.
If necessary, you can change it through the portal (portal.azure.com), but I just donβt see any reason to change it.
The title of the question is: "How to run the App Service in Azure as a domain user?" Is this really vital for you? If I'm not mistaken, then it is exactly the App Service that runs at all. This is PaaS (Plattform as a Service) and you manage there only your application, not the whole IIS farm. This is where the guys from the 51st minute talk about how App Services works.