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
I recently had horrendous problems trying to install Xamarin on VS2015 + Win10. (I don't have any other VS versions installed.) Even a full install and reinstall left huge holes in the installation. I tried repeatedly and each time the whole dev environment seemed to get itself in a different unusable state.
The fix for me, which is probably hitting a nut with a sledgehammer, was to:
This resulted in a faultless, fully functioning Xamarin environment for me. Overkill, maybe, but worth a go if all else fails.