Some cross-platform considerations have been built into it from the start, though. And there has been an official open-source implementation on a non-Windows platform. Perhaps more to show that it's possible than as an actual usable artifact, but it was there.

I think describing Mono as Official is a bit of a reach. AFAIK they benefited from the patent promise but that was about it until the acquisition.

If you mean .NET Core as official open source, that's true, but it's a relatively recent thing.

There was Rotor on FreeBSD. Probably more a proof of concept, but it was there.

Mono existed from very early on with .Net and the core was pretty compatible. I think when MS bought Xamarin, it finally got the attention it deserved, and really did go to help .Net Core take form.