In which month is your birthday?
In che mese è il tuo compleanno? In che mese compi gli anni?I am interested in the month, not in the date, so I need a specific question to highlight it, not the generic 'When is your birthday?
Why should I use 'what' instead of 'which'?

Question

In which month is your birthday?
In che mese è il tuo compleanno? In che mese compi gli anni?I am interested in the month, not in the date, so I need a specific question to highlight it, not the generic 'When is your birthday?
Why should I use 'what' instead of 'which'?

Answer

In what month is your birthday? What month is your birthday?

you can also ask  :

When is your birthday?
What date is your birthday?