So, 네 is like an abbreviation of 너의. 의 is used to show property kind of like the english 's. So that is fine to say, but as you mentioned, that is used in casual language or 반말. 당신 is a polite way to say you, so you could say 당신의 생일, but it might sound a bit stiff. A lot of times the word "you" is omitted since you can usually tell by context who's birthday you are asking. For example, 생일이 몇월 몇일이에요? If you are talking directly to one person, what are the odds you are asking your own birthday? So they often omit words like "you" or "I". Hope that helps.

Alanna Johnson's answer is perfect. Additionally, not 너 but also 당신 can sound rude. 당신 means 'you' in dictionary and it works in a book. but when you talk, it is usually used in uncomfortable situation (ex. serious argument ) if you ask birthday, say "생일이 언제에요?" (존대말) "생일이 언제야?"(반말) is frequently used than "생일이 몇월 몇일이에요?"