lol you guys were certainly talking about when you guys were born, not literally "age".. When I first read what you wrote I was like what the hack is going on hahaha Anyway, yes, I think she meant exactly that and it seems that her younger sibling was born in 1999. (동생 could be either brother or sister but certainly younger one) For the question part, I think you can simply say "편하게 언니라고 불러도 돼(요)?" which means "can I call you just (comfortably) unnie?" You might think calling her "unnie" is formal but it is not necessarily so. It feels very natural to Koreans to call a friend who is slightly older "형" "누나" "언니" and your friend might feel very comfortable talking to you.

'내 동생이랑 나이(가) 같다' - (You are) the same age as my younger brother [or sister]. "to clarify, are we both speaking informally or should i call you unnie?" => (서로 알고 있게 / 오해 없게 / etc), 우리 서로 말(을) 놓을까요, 아니면 내가 언니라고 부를까요? 말을 놓다 (or 반말 하다) means to do away with 존댓말 (formal, respectful speech) and talk informally.