Not in the slightest. Hacker spaces are an extension of just about every enthusiasts club ever, and if anything startup culture is a product of THOSE. Nobody sat down and said "you know, these startups are pretty cool, I should start a casual place to work on such a thing." Most hackerspace projects aren't even for profit, they're just things to learn on or group projects to socialize and expand the craft.

Remember that Wozniak tinkered with hardware for fun and his day job at HP. Such things were possible then, but before that there was always clubs for engineers and creatives of all kinds. We often ignore history when we create these kinds of articles, because writing things in the context of history is hard and doesn't get as many clicks if you don't put more effort into it. I find this phenomenon deeply interesting, as we'd do much better as a species if we learned historical contexts for a lot of things we think are new and powerful ideas.

tl;dr: No, startup culture is a niche extension of hackerspaces. Not the other way around.