'Birthdate vs birthday'- I know three other people who share my birthdate
Solution 1:
Like the bureaucratic, Date of Birth, birthdate includes the year. Baryshnikoff (Jan 27, 1948) and Mozart (Jan 27, 1756) share a birthday but not a birthdate.
I have met many people who were born on the same day of the year as me, but only one who also was born the same year. I did refer to her as having the same birthdate as me. (American English)