学校で教える標準文法だと必要でしょう。つまり、試験だとどこかにinを入れる必要がある。但し、口語では、What month were you born? で十分通じるし、ネイティブもそういう言い方をする。
What month were you born in?
Vicki Watson著 Grammar For Lifeからの引用
Participants were asked “What month were you born?” and “What year were you born” in order to create the continuous variable of age ...
Page Coyne他著 Examining the influence of CrossFit participation on body image, self-esteem, and eating behaviours among women からの引用
前置詞なんか使わずに以下のような言い方もよく使われる。
What's your birthday month?
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