While all of the projects listed above support VS Code extensions, only Microsoft products can use and connect to Microsoft’s Extension Marketplace. The terms of use for the Marketplace prevent any non-Visual Studio products from accessing it.
How does Cursor host extensions by Microsoft on Cursor’s Marketplace, such as C# and C/C++, without violating the VSCode Marketplace terms of service? The extensions are not accessible on OpenVSX either.
Disclaimer: I am not a lawyer. I was involved (a little) with getting Open VSX up.
Open VSX requires an extension author to sign up and publish the extension. It’s a manual process. However, it’s free to use and Cursor seems to integrate with it already (according to this discussion).
Technically, marketplaces are typically not “hosting” extensions they are linking to them. This is important from a legal perspective.
I don’t know what/how Cursor is doing it. However, based on other forum questions and answers it seems that there is some process which scans the VS Code marketplaces for new (and/or updated?) extensions and publishes them on the Cursor marketplace. The process maybe automated or run manually. I don’t know.
I think from a compliance perspective you can implement such a process/tool as a VS Code extension. This might not be in violation of the VS Code marketplace terms posted above because technically VS Code would be connecting to the marketplace. However, it may become complicated if some API is missing and the extension would need to initiate network connections to the marketplace directly. 
There could have also been negotiations with Microsoft to allow Cursor such synchronization (perhaps for a percentage of revenue). It’s possible as well. But I could not find any published information. Thus, just speculating here.
However, one thing that’s possibly a gray area is the implicit consent from extension authors. On Open VSX an extension author has to publish themselves. Thus, there is explicit action needed by an extension author to publish its extension. Cursor just seems to do it without asking/checking with the author. I think most authors would be fine with the implicit consent. But there might be a few which are not. That’s why I say it’s a gray area.
Thus, I recommend Cursor users carefully checking an extension terms and conditions. I believe some extension are not open source and only allow use within VS Code. I looked at this when considering a VS Code fork for a project. I decided against it basically it meant losing all remote development capabilities (their terms allow use within VS Code only).
Cursor uses OpenVSX. Unfortunately, a lot of extensions won’t show up.
This wasn’t too much of an issue before as you could download the .vsix from the official marketplace, but it looks like MS removed the download option, so you’re basically screwed unless the vsix is published on GitHub.
More info here: