Birthday paradox with a (huge) twist: Probability of sharing exact same date of birth with partner? I share the same birthdate as my boyfriend, same date but also same year, our births are seperated by merely 5 hours or so.
I know that the chances of meeting someone who was born on the same date than me is fairly high and I know a few people with whom I share my birthday although for the little I've read about the birthday paradox, it doesn't take same year into account.  We've argued before about the probabilities and I am still not satisfied. My point was that the chances are tiny if you consider the probabilities of being in a relationship (+ being successful at it for X amount of time). I find the amount of factors to take into account quite vast (up to a point, gender and age, availability, probabilities of separation in our region, etc.)
Is it even possible to calculate the probabilities on something like this? How would you go about it?
 A: If it's an event specified before the fact, you can simply break it down:
The chance that your boyfriend was born the same year as you is actually very high (especially given many situations tend to bring people of very similar age together); it's a very difficult probability to calculate, though, without data. 
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).
So if you had some good estimate of P(Same year), you can calculate the overall probability reasonably well.
I'd guess that P(same year) is roughly of the order of 0.1 to 0.2, but that's just a guess. [Edit: Jeromy gives a figure based on actual data, which turns out to be about 17%.]
A: Taking the question literally
According to wikipedia, 33.2% of married couples in the United States differ in age by less than one year.
Thus, a baseline estimate for sharing the same date of birth would be the above statistic divided by two (because it captures 2 years) for sharing the same year multiplied by the probability of sharing the same birthday:
$$P(DOB_i=DOB_j)\approx \frac{0.332}{2} \times \frac{1}{365} = 0.00045$$
Or roughly 1 in 2200.
As has been noted, both the shared year and shared birthday probabilities could be further refined based on additional information.


*

*Probability of shared year: The distribution of age differences in relationships varies based on many factors including culture and time. Also, the above statistic is for a year difference. Dividing by two might lead to an underestimation because the probability of being within six months of age is probably more than half the probability of being within a year. 

*Shared date of birth: There could be tweaks to this. In particular, the uneven distribution of births throughout the year could have a small effect. If you're born in a leap year, then you have 366 as the base divisor. Then there is the elusive effect that being born on the same day might have on you both. In particular, if you are a person who reads into such things and searches for coincidences, such a coincidence might subtly increase your chances of staying together. 


Thinking about other coincidences
When it comes to two people, there are many potential sources of coincidences. Humans are very good at identifying patterns. Within the domain of date of births, you could imagine many possible similarities: same month; same day of month; same star sign; same birthday, different year; some similarity in the numbers such as 2nd of May and 5th of February; dates are some round number apart (e.g., 8th of May 18th of May); dates are only only some small number apart (e.g., 8th and 9th of May). We could in some sense describe our sense to which any of these feel surprising or like a major coincidence.
But of course, when we speak about coincidences there is a much wide domain of search. For example, we could look at similarities in names, employment history, appearance, etc. The larger you cast the search, the more possible bases there are for finding coincidences. 
In general, the more you look for them, the more you will see them. This is analogous to the analyst who performs many post-hoc statistical tests without correcting alpha. With enough analyses, the probability of finding a significant pattern gets close to one even when alpha is small.
A: 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. If you add in year, it's probably something like 1 in 3000 or 4000 (most people have relationships with people relatively close in age).
But that' a priori. 
That is, if you had asked, before meeting your current boyfriend "What are the odds that the next man I have a relationship with will be born on same day and year?" the odds would have been 1 in 3000 or so.
However, post hoc (that is, while in the relationship) it's trickier because you would have noticed a lot of other coincidences too: My boyfriend was born the day before me!  My boyfriend's mother has the same name as my mother!" etc etc.
The odds of "some weird connection with my boyfriend" are impossible to calculate. 
A: As Peter pointed out, it is impossible to calculate coincidences after the fact.
Your question got me thinking, and I realized my girlfriend and I also have a strange birthday coincidence.  She was born exactly 432 days before me!  And we are also in a successful relationship!
I don't know what this probability is, but it is the exact same as yours!
A: The chances for this to happen.... two people having their birthday on the same day as explained by the other posters is 1/365 * 1/30 to be conservative here with the age ranges. To be in a relationship, a successful one multiply by maybe 1/2 or 1/3?! 
However, for you to be in a relationship, you first have to be here. For you to be here, your mom and dad needed to get together - how likely was that then? Then their parents, grandparents, great grandparents, predecessors, apes, fish, amoebas, rays of sun hitting the first predecessors to plants, back to the big bang going as it did and whatever was before it. If you consider all, then every atom in the universe had to be exactly the way it was for you to be there. 
You could almost say it's a miracle you guys got together.
A: Although the question is about birthdays, the "birthday paradox" isn't really relevant here. It's about how many random samples you need to take before you expect at least two samples among them to be equal (a collision). Your question is mostly about the probability of two samples being equal. If there were 30 people in your relationship then you'd expect two of them to share a birthday but there aren't 30 people, there are only 2.
The odds of having a relationship only have quite a small effect. Most people have a relationship at one time or another. I'd guess more than half of adults have one right at this moment. Some people have several at once, mentioning no Présidents de la République  in particular ;-) So it's not going to massively reduce the odds, maybe halve them.
The main consideration is, given this significant person, what is the probability of them sharing your birthday? On pure chance it'd be roughly 1/365 given that the person in question exists at all. Since you choose a partner based on everything you know about them, which includes their birthday, you can't discount the possibility that the actual incidence is significantly higher or lower.
Look at it another way: what's the chance of there being someone who delivers your post and shares your birthday? The chance of a randomly-selected person being the one who delivers your post is tiny, but so what if it is? It doesn't affect the answer. Assuming universal delivery (which I can in my country), someone delivers my post. If there's only one then the answer is roughly 1/365. We can completely exclude from consideration all the people who don't deliver my post, they don't affect the odds no matter how many of them there are.
What's the chance that you have a partner who shares your birthday? It's about 1/365, times the chance you have a partner. Then adjusted by any factors that mean sharing your birthday is correlated or anti-correlated with dating you.
What's the chance that your boyfriend shares your birthday? Well the question pretty much assumes that you have a boyfriend, so strike that part from consideration!
To incorporate the year you need to look at the way the age differences in relationships are distributed. As a rough guess, I'd look at what proportion of relationships have an age difference of less than a year, and multiply my previous number by that. Of course, if you have access to that kind of data you might just be able to look at what proportion of relationships match your criteria, and get the exact frequency without estimating anything :-)
In a society where there's a strong tradition that the man should be somewhat older than the woman in a relationship, you might find that the proportion of age differences below a year is very small, and the proportion couples who share date and year of birth is tiny. This could be the case even if the average age difference is just a couple of years. So maybe you are special, by bucking society's rules. Myself, I'd guess that the proportion of relationships with an age difference less than a year is probably over 10%. But I wouldn't be surprised to be wrong and besides, a lot of my friends met their partners at university, which clearly affects the age difference among the available candidates and that biases what I see to make my guess. Everyone's equal in modern society (right?), but "the man a couple of years older than the woman" is probably a stereotype for a reason.
A: So, first of all, the odds of sharing some weird connection with any random person are probably quite high. From experience I'd guess around 20% or so, no way to seriously calculate that, but no matter what it exactly is, just want to be clear having a special weird connection means nothing (though it is fun).
Then, something the other didn't take into account, looking at birth rates per month

we get a nice overview (it's caused by things like people being off thus having a lot of free time on hand 9 months before the months in question), next dividing that percentage by the number of days in that month.
Next one should figure out what the chance is of being born in the same year. Just to give an impression of this chance I'd start with the rule of thumb as presented by xkcd which gives a dating age range of $\pm \ age-\frac{age}{2}+7$, where age $=$ age you started dating. Which in turn gives $P(same\  year)=\frac{1}{pool\ size}$. However, the chances of dating someone from the exact same year are far greater due to a flaw in the educational system where pooling is done by the date of conception. A consequence of this is that the pool of people you know from the exact same ages is by guesstimation a factor of 5 bigger than the expected value, if $age<24$ or so.
What that exactly boils down to depends on age and month born, but for me it boiled down to more than 0.2% (1 in 500). Definitely not normal, but then again, coming full circle, you will find something like that for everyone after the fact. 
