It's not, because it's being applied outside of its defining group. It's a circular definition if the term is only applied to "start-ups", because of course those have "start-up culture" by definition. But one can apply the term to companies that are not "start-ups" -- even if they once were -- but yet carry the same "start-up culture".
There's also the idea at this point of a stereotypical "start-up culture" that lives beyond the idea of whatever a "start-up" is. This also breaks the circular definition, because from that perspective "start-ups" no longer define "start-up culture".
