Question

is there any day of a year, in which anybody's birthday does not falls?
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Answer 1

Everyday people are born, and everyday people die. Yes. Days which occurred before life existed were ones during which nobody was born. More narrowly speaking, days which occurred before people came into existence (“nobody born” as distinct from “nothing born”).

Answer 2

yes ,any day of a year, in which anybody's birthday does not falls.

 Would it make sense to say that your birthday was on a different day, or to say that over the course of 4 years an extra day was added to the calendar year to try and stop the seasons from drifting.

Either way the birthday following a leap day will be a day further away than it normally is. If someone's birthday fell on the 29th Feb, then they will have had to wait only 365 days from this year till next, however th birthday following that will be 366 days away. It's the same for everyone no matter when your birthday is so that's why we don't complicate things.