The Azure SDK is what makes it possible to manage resources in one or more Azure subscriptions from within Visual Studio. You'll find it in the Server Explorer next to the Data Connections, SharePoint Connections, … In the past it has always been possible to connect using a Microsoft Account or Organizational Account, but it looks like something changed overnight.
It looks like you can't use a Microsoft Account anymore when connecting to Azure from within Visual Studio:
This is fine if you have an organizational account, but if that's not the case you might want to follow the next steps to keep using your Microsoft Account. Start by clicking the link Can't access your account?:
You'll be redirected to the password reset page. Press Cancel to return to the login page.
You'll end up on a different login page where you can connect with a saved account (including your Microsoft Account) or choose Use another account.
The Use another account option will allow you to choose any Microsoft Account or Organizational Account:
Enjoy!