No, Core happened before Mono integrated in to the Standard platform. I believe Mono started to use the open sourced components of Core/.Net framework to replace some of their components.That might be where the confusion lies. Then at some point after that, Microsoft acquired Xamarin and opened the Mono/Xamarin platforms fully (the Mono runtime used to have tricky licensing that meant that embedding it required a commercial license.)

.NET Core is incompatible with the .NET Framework which Mono is the oss counterpart to.

That's the official line but I honestly don't know anymore. We were trying to write a few cross platform libraries that we could also consume in Framework. We could not get .NET Standard libraries to import into Framework Apps without entering a dependency hell that made me long for the DLL Hell of the 90s. We discovered that re-targeting .NET Standard projects to .NET Core would allow them to be imported into .NET Framework projects without issue.

Of course you can't import .NET Core libraries into .NET Standard so in the end we actually made projects for each of them with links back to the .NET Standard libraries' files.

I think it depends on what you are trying to achieve. They are all slightly incompatible to some degree. I've written quire a few libs in .Net Standard and had them work on all three platforms, as well as writing Core command line apps and ported them to .Net 4.5. The issue is more with stuff like ASP.Net Core etc, where the libraries are slightly different conceptually.