It's like saying, on the date I was born rather than in the date I was born. :3
I'm getting a pony on my birthday vs. I'm getting a pony in my birthday.
- I'm turning sixteen on my birthday.
- I really want to go out of town on my birthday.
You can typically just tell from how it sounds. :)
I suppose you use "in my birthday" sometimes, when you're not referring to a date and more like an adjective.
- My mom put pudding in my birthday cake.
- I hope I get money in my birthday card.
Just think of it like, what kind of cake is it? A birthday cake, so you use "in", or what kind of card is it? A birthday card, so you'd use in. Otherwise, if you're talking about a date, you'd use "on".
I'm sure there are exceptions, but that's generally how it works. c: