Yes. Perhaps not always in the way you are thinking, but yes, absolutely.

I'll focus on rural Christian communities here, but Jewish communities (primarily urban) and Muslim ones would have followed parallel patterns.

First, days of the week. Above all, knowledge here would have been governed by Sunday.

The only church attendance requirement was that the Fourth Lateran Council in 1215 mandated presence at church twice a year: once during Lent to say confession and receive absolution, and on Easter Sunday to receive the Eucharist. In the fifteenth century, some dioceses would require additional reception of confession at other points in the year.

However, it's very obvious from surviving sources that people vastly exceeded these requirements. Medieval church services weren't people sitting quietly in pews and kneeling on kneelers with folded hands at appropriate moments. Lay people segregated by sex jumbled together, standing and walking around, and above all talking. The monastic demon Titivillus, who gathered up loose language during the liturgy and spat it back out at gossipy monks on Judgment Day, made appearances in preaching to the laity as well. Going to church on Sundays and feast days in particular was the center of village/parish social life.

Second, date of the year. Medieval communities had two calendars: the month-date we think of by default, and the liturgical (religious) calendar of the particular faith group. That means liturgical seasons and holidays/feasts. So: Lent, Easter, Pentecost, Ordinary Time, Advent, and so forth as seasons; All Saints' Day, Michaelmas, St. Andrew's Day, &c as holidays.

This is primarily how medieval people figured dates. Even in the early Reformation, Protestants date their letters according to "the second Thursday before St. Katherine's Day" and so forth. Church attendance and the different rules for behavior on feast days and solemnities meant people kept track of the liturgical date.

(Sidebar: until the spread of mechanical clocks in cities in the 14th century, and later in rural areas, time was also calculated liturgically--according to the standard times for monastic prayer. Hence "o'clock": the time of the clock, not the monastery.)

Third, year. Anno Domini became more and more commonly used over just this period in question. In court testimony by peasants, however, many people continued to date the year with reference to a major event in local or national life, such as a king's coronation. It was a different type of relative year, but court records seem to suggest an accurate one.

As far as birthdays, that is a tougher one to prove or disprove for peasants. We do have scattershot surviving parish baptism records (with untold more lost over the centuries), so there was at least record of that significant date. And baptism was a HUGE deal in the Middle Ages and early modern era--baptism marked the baby's entrance into the community, but it was also a community event overall, especially with the union of the parents and godparents into essentially one family.

Overall, peasants in the later Middle Ages unquestionably understood their situation in time.