It's "on your birthday" because we always use the preposition "on" with days.
It's "on your birthday". It's incorrect to say "in your birthday" because someone's birthday refers to a particular date, and we always use the preposition "on" with days.
We went out for a nice dinner on her birthday.
We went out for a nice dinner in her birthday.
Follow the same logic when using a specific date.
His birthday is on March 6th.
When using the determiner "last" or "next", however, we usually omit the preposition "on".
My birthday is next Friday.
When do we use the preposition "in"? We say that someone's birthday is in a particular month.
My birthday is in July.
My birthday is on July.