"Are you born in July?"
Translation:Er du født i juli?
Yes, the English translation doesn't make sense at all.
As for the Danish sentence, it's in line with the way you would say this in French and German and probably in other languages, e.g.
Je suis né en juillet. Ich bin in Juli geboren.
It does sound like you are saying "I am being born", but you should think of it more as a present condition based on a past event sort of like the present perfect tense.
Example, when you say "The car has been washed", you are really saying two things: 1- the car was washed at some point in the recent past and 2- the car is now clean. But, as soon as you spill coffee in the car or it rains on the car, it's no longer true that the car has been washed. Now it had been washed but is no longer clean.
If that explanation is confusing, just consider that languages work differently. Just because we say "I was born", it doesn't mean other languages work the same way.