For consistent branding around all tools and technologies that could feasibly be used to build applications on top of Azure

That's not a super strong argument.. Visual Studio -> Azure Code Studio, TypeScript -> AzureScript, Windows -> Azuredows.. (Only a little tongue in cheek )

That's still not good naming: Azure has so much services you could practically run anything on it. If keeping the Azure name to products actually running in the cloud it makes sense, otherwise it's just brand dilution.

Originally it was built to manage SQL server instances on Azure.