Where did the celebrations of a birthday originate from?

I suggest you get at the root of the issue.

Who tells you on which day the birthday falls and on what basis?

It essentially presupposes recking of time and that is an issue in Astronomy. We both being Astronomers I wonder what you are doing in this section.

Birthday may mean recurrence of same day as per Solar (western) calender, Lunar calender correct to comply with Solar calender (Indian traditional system), Lunar calender (pure version as in Islamic calender).

Culturally candles, birthday song, and so on were started in some culture. In a course of history, much more globalised for the last two centuries, it was marked by cultural domination (or its reverse process of cultural aping) of the ruling groups over the native subjects that is seen glaringly. So people simply adapted it and it has come to pass off as a native custom. Other ways of celebrating the same are there in other cultures too, some observed in their purity and others in a mixed-up manner. The latest generation now is confused, unable to separate these separate cultural strands. You can draw your own conclusions.