Facebook culture makes a similar statement: "Nothing at Facebook is someone else's problem." I don't take it to mean that you are supposed to be the Atlas of the organization, but rather that when you see problems and issues that might arise, you don't ignore them because it's "not my job" or "not my problem" - you embrace it, escalate or forward it to the correct people, and then go back to what else you have to do.
This type of thinking stops issues from being buried when they're noticed by someone, even if it's something outside of what they are judged on in their responsibilities (and performance reviews). A good hypothetical example of this is a crash condition you may trigger as an engineer. You might not quite know what is going on but you have a reproducible testcase of a crash condition in someone else's stack, and saying it's not your job means the bug doesn't get fixed or reported. Had you submitted that crash condition and made it temporarily your problem, you could have indirectly helped patch a deserialization bug that could've led to code execution on that tier with a more malicious testcase (i.e. exploit).
In smaller companies, I think everything technical kind of ends up being your job, so your job becomes what you make of it at that moment. Do what needs to be done to ship the thing.