같다 - same or very similar; 똑같다 - exactly the same
yet "exactly the same" isn't an accepted answer here
I would argue 같다 is "similar" (not same) and 똑같다 is "same".
"Exactly the same" would require a similar modifier like 완전히 or 굉장히 or something.
같다 = alike, similar (but not the same one.) ~ doubles
똑같다 = be exactly alike, identical (but not the same one.) ~ clones
바로 같은 or 정확히 똑같은 = the very same (one and the same)
Why not 우리의
It's more like "we have the same birthday" rather than "our birthday is the same".
think of it as "as for us, the birthdays are the same." korean language constructs don't map one-to-one to English
"We share the same birthday" is another way to put it.
Would this imply the year as well, or just the day? Or would that be based more on context/clarification in the situation?
Like in English, 생일 ( =birthday ) is only about day & month.
It is not the same as 생년월일 ( = Date of birth ) which includes the year of birth.