first of all, men and women are different but i think that your moon sign, rising sign, etc.
I am dating a man who’s birthday is the same day, same month and same year as mine. I would be bored to tears dating someone so similar to myself.
If you had that probability, P(Same day and same year) = P(Same year) $\times$ P(Same day|same year).
But P(same day) should be roughly independent of whether you were born in the same year.
So it will be $\approx$ P(Same year) $\times$ P(Same day).
For any one relationship, the odds of sharing the same month and day are approximately 1 in 365 (not exactly because of leap year and because births are not exactly evenly spaced within a year.