I have Xamarin installed on the Visual Studio 2013 on Windows Operating System. It works fine.
There is Xamarin tools in VS2013 as follows:
My problem
We have a few troubleshooting steps on our Knowledge Base that you can find here:
https://kb.xamarin.com/customer/en/portal/articles/1643349-missing-visual-studio-extensions-after-installation
Note: Some of these steps have older versions listed in the respective paths. You should use the latest version number available.