Microsoft's modern documentation website is produced largely from Markdown files hosted in on GitHub. Typically the documentation shows a left pane navigation menu and a document navigation section on the right. Here is an example:
https://learn.microsoft.com/en-gb/dotnet/fsharp/what-is-fsharp
Microsoft also produces a set of tools to help documentation authors and associated extensions for VS Code.
Is it possible to use this to host Microsoft documentation inside your own website? If so, are there any examples of doing this in an ASP.Net Core website?
You might want to have a look at this blog post of one of the Microsoft employees working on Docs, describing on how to set up and use DocFX, which Microsoft Docs is based on.
Also, this GitHub issue gives a good summary of the current status. (All issues in the linked repository have been deleted.)
Aside from that, the links you mentioned correspond to this Visual Studio Code extension package, where you might be especially in Docs Preview. It will help with writing and previewing Docs Markdown flavor syntax.
The repository links back to the packages GitHub page though, the source code for Docs Preview seems to be unavailable and also the license is not permissive at all, only granting Microsoft rights, but not granting you any in return. Even though you could look into the code of the downloaded extension manually, that would not grant you any permission to use any of the acquired code or knowledge in any commercial project.