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.