.NET CoreCLR is indeed open source, but I doubt you can use it for Android or iOS.
Unity3d has been using Xamarin's .Net to build for Android and iOS (and many other platforms) for years now.
On iOS the situation is even worse you will need to have an AOT compiler for C# in order to be able to ship iOS apps in the Apple Store.
You are making a confusion here. Unity doesn't use Xamarin at all, they have an old license (like really old, since before Xamarin even existed) for Mono for iOS and Android.