Do you know the song "hashkediyah porachat" which everyone sings on Tu Bishvat? How can you not know it? How can you not be sick of it? Anyway, it says that every year in Israel, the almond trees begin blooming around Tu Bishevat.
That's a pretty useless piece of information if, like me, you have never seen an almond tree in your life. But if you do have almond trees nearby, it just might come in handy. Imagine you're an almond farmer. Also imagine that it's January 27, the 20th of Shevat - but you don't have a calendar, so you don't know this. Somebody comes by and asks you what the date is. You can't just look at the calendar and tell them. So you look out the window and do some quick calculations. Your almond trees blossomed roughly a week ago... and that usually happens around Tu Bishevat... so it is almost certainly Shevat now. Maybe Adar, but probably Shevat.
Shevat is not the only month which is easily identified biologically. The Torah describes Moshe's sending the spies as being in "the days of the first fruits of the grapes". It does this as background, because the spies bring back grapes with them. But anyone living in Israel knows that grapes first grow around Tisha Beav, so that must have been when the spies story took place.
In fact, every crop has particular dates on which it sprouts, blossoms, forms fruit, dries up, is harvested, and so on. And the cycles of heat and cold, and rain and dryness are predictable too. So a thoughtful observer, without seeing a calendar, can almost always make a very good guess as to what the month is.
What about the day of the month? Here the calendar's lunar aspect becomes indispensable. Look up in the night sky, and you can immediately tell what day of the month it is, more or less. New moons occur on the first of the month, full moons on the 15th. In-between phases correspond to other days. Also, though I have not seen this mentioned anywhere, the time at which the moon rises varies predictably with the day of the month. So by looking at the moon's position at sunrise/sunset, you could get a more accurate estimate of the day of the month. Overall, I think you can consistently estimate the day of the month to within one day, maybe two.
Going by the fruit-and-moon method, how accurate would your dating be? There's a good chance you would be off by a day or two, and a smaller chance you'd be off by an entire month. But much of the time, you could exactly guess the date simply by looking at the natural signs around you.
Such precision is not possible with either a solar or a lunar calendar, only with a mixed calendar such as the Jewish one. I don't know what other cultures did before urbanization and mass literacy allowed everyone to know the date. Perhaps they were just ignorant of it. In any case, when the prophets wrote in the Bible that such and such happened on such and such a date, I'm sure they didn't check their cell phone displays before recording whatever happened. Perhaps they consulted written records, if available. Or perhaps they just went into the field, and used natural signals to reach the exact same conclusion.