Yes people in the 16th century were aware of the year and their calendar.
The average person might have considered it less significant than the average person today, due to the fact that literacy was not as big a thing and that far more people were engaged in agriculture than today.
However the 1500s was also when the current Gregorian calendar was invented to replace the old Julian one. Since that invention came from the Pope and lots of people were heavily invested in not being catholic, the exact date actually became a thing people care very much about in those days.
The 16th century was also a good time for prophets of doom who were presaging the end of the world and the beginning of the end times. Knowing exactly which year it was, was required for knowing how much time was left. Lots of these apocalyptic scenarios were based on numerology and astrology and the fact that 15 centuries had passed since Jesus birth/death featured in a number of them.
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