Does anyone know of a game engine that uses CoreCLR instead of Mono (like in Unity)?
For something more professional and polished, there is Xenko, which runs on CoreCLR for some of its non-mobile targets. https://xenko.com/
Also, Unity itself doesn't really run on Mono -- it just has a scripting layer that hosts Mono on top of its C++ core. Perhaps not a useful distinction for you, but I thought I'd mention it.
(Note: I work on .NET Core at Microsoft)
At the same time the C#/.net programs are executed by the Common Language Runtime.