State ownership is merely one interpretation of the base communist concept of communal ownership. The internet is largely run on protocols built organically by the people using them and only later codified into standards by a committee of interested parties.
Pretty sure that's socialism. Communism is envisioned as a stateless society. Which is kind of what the internet is. There's standards bodies, but they don't really have any enforcement power. And ICANN, one of the few bodies with actual power, (by controlling domain name distribution) leads to problems like in this article.