What does it mean if you and your partner have the same birthday?

Unless you were born at the exact same time in the exact same place, your rising sign (if not your Mercury sign as well) wont match your partners. It simply implies that you and your partner didnt instantly connect on every astrological level.

What is it called when you have the same birthday as someone?

The birthday paradox, also known as the birthday problem, states that in a random group of 23 people, there is about a 50 percent chance that two people have the same birthday.