Hey, I will try to answer the question using one simple example and then escalating it onto a broader scale. Please keep in mind that I am mostly talking about Early Modern Europe. For other times or continents other more knowledgeable people will have to chime in.

Your question brings with it one simple yet kind of huge problem. How are we supposed to find out, if the average person knew what year it was (common sources are rare)? Especially considering that different cultures or realms counted years differently. Considering ancient times, the Roman Empire used the Consuls as their counting measure. The Year depended on the Consuls of the city. It is also common (in Imperial times) to find years counted relative to the rising of a certain Emperor (e.g. "three years after the appointment of Emperor Traian"). It is important to note that Romans counted inclusive, meaning that "3 years after..." would mean "100 a.d., even though we (in the year 2020 in most of the world) would count 98, 99, 100, 101(!) - 101 a.d. being 3 years after 98 a.d.

This small piece of fun fact aside, the question still stands. How do we know what average people knew about the current year? Most of them didn't write, many of them couldn't read. To still try and answer your question I will consider one specific source from the early 17th century:

The farmer and shoemaker Hans Heberle, who lived in a small village near Ulm in the south Germany, wrote a diary. This diary is a pretty important piece of information for our knowledge of specific economic questions concerning Ulm in the Thirty Years War. But it also shows us other things, especially considering your question of keeping track of current years. He knew what year it was. He counted the years and days (albeit in the protestant format, not the Gregorian one we use today - that means to get "our" dates we have to add roughly 10 days to Heberles dates). And they are, to our knowledge, really accurate and make perfect sense if we compare his dates to other unconnected sources of the time.

This sounds cool right? A farmer kept track of dates and years! Well, sadly this is not that great news for your question as it seems on first glance. Heberle knew a lot. Kind of too much already. He stands out in many ways. He knew of specific troup maneuvers/sieges and about what happened in other parts of the Holy Roman Empire. He was well read and rather well written as well. Comparing him to other sources of the time he got a lot of his knowledge out of "Flugblätter"/"Flugschriften" - small news carrying (printed!) pieces of paper. Often using words or pictures to inform bigger masses of events, kind of like a heavily moderated news page. So he cared about the events that unfolded in the Holy Roman Empire and he wrote them down.

So what does all of this mean for your question? We have 'common' sources (of average people) that tell us exact dates and these dates match the dates named in other - unconnected - sources (military letters, church books, etc.). But common sources are rare the further back in time we go. Sadly. So we do know there was at least one person in this village near Ulm and considering the fact that Early Modern Society was face-to-face (everybody knew everybody in their small village) it seems reasonable to assume that this knowledge was shared throughout the Village. Cities are an entirely different question, but since your question seemed more interested in farming people I tried to go into depth there.

So, to conclude, we know of "average people" in the early 17th century that knew how to keep track of years and dates. Extrapolating knowledge we have on rural society as a whole it is reasonable to assume that "a village" roughly knew the year and date.

Edit: some spelling errors

As others have pointed out, the question itself presents certain problems. For one, the system of a single, linear, continuous dating system is by no means common across the globe before the advent of globalization takes the Christian formulation to be the standard. In many places, years were marked by the reign of the current ruler, with the count starting over each time a new one took power. Additionally, even in places where a timekeeping system like ours was used, the average person might only have an approximation of, for example, the year they were born. In some cases, they might have an accurate yardstick by a grand event that took place in the year of the birth.

But to get more directly to your question, I can speak somewhat to the situation in the United States (and its forebears). Accurate tracking of the date (although we should bear in mind that words like 'accurate' make it seem like there's an objective measure of time being used, which there is not. It's totally arbitrary as far as the grand scheme of time goes.) would vary widely across demographics and locales. The American colonies were heavily varied.

And, as you'll see, things like knowing the date are connected to a bunch of other cultural practices that either make that possible or make it useful: people do not develop systems to that don't serve them in some way. So knowing the date is all fine and well but it's a waste of time and energy to even bother with it if it doesn't make life easier or better. So what you'll see is that things like date keeping are tied to literacy which is sometimes tied to religion and sometimes tied to money and sometimes tied to politics, etc.

In, for example, what's known as "Puritan" New England, it would be relatively likely that the exact date was known by virtually everyone of school age. Congregationalists, the more accurate name to describe the family of religious movements that includes the people referred to as Puritans, heavily valued literacy, in part so that every person could for themselves read the Bible. In practice, this meant they would see schooling through to at least about 8th grade equivalents, giving them much exposure to the date. They would also experience heavy social pressure to attend church services every week.

Other religious groups that valued education, like Quakers in Pennsylvania, Lower and Western New York, and New Jersey (and later much of the Midwest and the tippy-top of the South), also experienced high rates of literacy, and would have been likely to know the date.

In some areas of the colonies, however, like the Carolinas, people generally lived in very sparsely populated areas. Their lives would thus revolve around the rhythms of farm and small-town life. The people most likely to know the date likely lived a ways off and did not regularly communicate with the average family, e.g. a government official above the municipality level or a judge. Very few people in this area would attend formal religious services more than a few times in their lives, with many only seeing a pastor once every few months, when the itinerant preacher made his rounds through their area. Literacy was, for obvious reasons, relatively unvalued: what import could books hold in their real, everyday lives?

In other areas of the South, you'd be likely to find a wealth-segregated spread. Virginia, for example, was home to a great many wealthy plantation owners and larger cities than the rest of the South. Here wealthy aristocratic families would hire tutors to teach their children in the home, and the kids would likely be expected to know important historical events, to be fully literate, etc. Again, this would make them more likely to be regularly apprised of the date, especially depending on how involved their father was in local politics.

Moving into the Young Republic Era, the US was still more farms than anything else, but letter writing is becoming more popular as the USPS becomes an office of the government (and later part of the Cabinet!). This facilitates knowing the date because people might date the letters and commonly needed to have a common referent in time to explain events to their interlocutor. You also have the introduction of regular elections, i.e. ones that always occur at set intervals. As we saw before, these kinds of shared events make it much easier for people to keep time across distances.

With all that being said, it's hard to say in a lot of history what the "average person" knew or did not know because the average person didn't write: even if they could, paper and writing implements have been expensive for most of their existence, even beyond the privileged status writing has often had.

Diaries can be super helpful in this regard, and I do recommend checking out A Midwife's Tale by Laurel Thatcher Ulrich, a classic of microhistories. It takes as its primary primary source the diary of a New England midwife named Martha Ballard. Now, in some ways we can take Ballard to be a regular-Jane, though she did have some informal medical training. And she kept very meticulous records of her days-in and days-out with precise dates. And one of Ulrich's masterful points in my opinion is that people's, perhaps especially women's, lives were marked by relentless dailiness, filled with mundanities and repetitions and boring details. But Ballard found them worth writing about, and the date worth knowing.

In all, I would say there are parts of the US where most people would've known the date most or all of the time straight from the beginning of colonization by white Europeans. There are others where it's unlikely people regularly knew the date until perhaps as late as Reconstruction. The hard part is that the places where it's hardest to tell are hard to tell precisely because there are no sources from "regular" people.