"My birthday is in July."
Translation:Mijn verjaardag is in juli.
It's not wrong. However, even though it's a synonym for 'verjaardag' (which we use for 'birthday'), 'geboortedag' is said to be referring to the actual day of birth. So, the day when you were born, and not a year later, or 20 years later. Just that one day.
Would "birth date" also mean this? I've wondered this before. It seems like it is but if you asked someone their birth date they think you meant birth day.
The same way that on is wrong in English. An event is "On" a day and "In" a week month or year.