Are people with same birthday compatible?

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.

How rare is it to meet someone with the same birthday as you?

a 1/365 One person has a 1/365 chance of meeting someone with the same birthday. Two people have a 1/183 chance of meeting someone with the same birthday. But! Those two people might also have the same birthday, right, so you have to add odds of 1/365 for that.