It's not a straightforward example of capitalism because it doesn't earn enough through its service to support itself, and must be propped up by an otherwise external entity.
The concept of "private ownership" is not definitionally identical to the concept of "capitalism", I hope you realize...
Private ownership and the protection of those property rights are exactly what capitalism is. As long as there is no coercive entity dictating what YC can and cannot do with it's capital then HN absolutely is a product of capitalism.
Capitalism doesn't mean "earn[ing] enough through its service to support itself". Why do you think that people are not free to finance sites that promote other branches of their business under capitalism?
That's an interesting question. It's true that HN doesn't earn revenue and that it could not stand on its own. But it's also true that YC is more valuable with HN than without it. I think you have to mark that in capitalism's column, even if it's a weird edge case.
Maybe more companies should be more open to more weird edge cases and maybe that would make them richer and therefore more capitalist? I don't know.