I don't think that ALL of them had 360 days, but for many of the older cultures in the middle east (until finger-counters took over), base 12 was used, and by derivative, 60, and 360 were common units. 360 degrees in a circle, 360 days.
Oops, shouldn't have said all.
I guess I don't follow. Where does the 360 degrees in a circle come from again? How did these cultures attain base 12 counting? I haven't run into any explanations that make sense yet, perhaps you have.
If they could do something as sophisticated as count in base 12, surely they could count how many days in a year there were. Being off by 5 days/year means that within a short time, their seasons go out of whack and their harvest gets affected.