The key word in "birthday" is "birth". Not conception.
For human cultural purposes, you are 0 days old at the time and date recorded for your birth. It goes on your birth certificate. If you find a culture that celebrates conceptiondays and issues conception certificates, let me know.
The comment I’m replying said nothing about “birthday”. They asked how many years old you are when you’re born, which is ambiguous.
You know some cultures don’t even celebrate birthdays, right? It’s not even uncommon for people to not know their date of birth. Even my own grandmother, born in Chicago in the 1920s, didn’t even know exactly what year she was born in.
When you're born. That is your birth day, because the person to whom you were born gave birth to you that day.
No culture I know of tracks the date of your conception as a measure of your age. If they measure it at all, the starting point is your birth.
You are zero days old on the day of your birth. Even if those cultures think you are "in the first year of your birth" for days 0-365 of your existence, they would accept that 1 day after your birth, you are 1 day old, not 9 months old.
Typically 3 months shy of 1 year, so about 0.75.