So I was reading a discussion somewhere (perhaps here) where someone was explaining that "web 2.0" was the "social web". Then it struck me that a lot of people don't even remember when this term was introduced, so to them this sounds like a very likely explanation.
The Web 2.0 term was introduced to differentiate the old page based browsing where everything was links and form submissions, to the dynamic web pages based on XMLHTTPRequest, allowing developers to write actual applications on the web. This is referred to as SPA (single page application) these days.
Social media didn't have anything to do with it, and it existed before the introduction of web 2.0. Of course after SPA's became a thing, social media sites started doing it as well, but it wasn't dependent on it. Early social media sites definitely didn't have any SPA functionality.