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.
"Startup culture" is purely an invention of "How do I build a business with no resources"? Hackerspaces are the continuation of HAM radio clubs and electronics clubs and engineers bullshitting at bars and people letting folks goof around with their broken television sets and kids goofing around on BBS.
None of that has anything to do with startups--to say otherwise is pretty much just astroturfing.
Formal hackerspaces (e.g. have a building, have staff, have policies) are [potentially non-profit] startups themselves.
Yet the extension of startup culture is true in a sense of massive participant demographic overlap. Mostly white, highly educated, male, generally hands-off technical background, very young and inexperienced, generally far above median income, overachiever in academic or vocational or artistic or perhaps all three, anti-individualist because its a socializing space, overwhelmingly urban extroverts... A bunch of socializing young java programmers aspiring to learn how to solder could be a startup staff or could be a hackerspace, depending on where they're sitting and who's paying the rent at that instant.