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While calendars and timekeeping were important to many cultures throughout history, they were usually for reasons relating to weather and seasons. It didn’t really matter how old you were, and few people would care exactly which day they were born on. u/Atramhasis writes that birthdays didn’t really become a thing in ancient Persia and Greece until astrology became popular:

So to answer your question directly, this is the point where more and more people would have been interested in the exact day of their birth. Astrology uses the position of the planets in the zodiac on your day of birth most prominently, and so if you wanted to get a horoscope that was accurate you needed to know your exact birthday. I would imagine there were a lot of people who would make up a day and not be certain, but often the way you would even know your birthday was that your parents would have your horoscope cast shortly after you were born. Prior to the “democratization” in a way of astrology, I expect you are correct that very few if any individuals felt the need to keep track of their specific birthday, and would be content only to know the passage of months and years.