Because this is for masculine nouns.
Eu sunt al doilea băiet.
Fata...no? We use the definitive...or no?
No. In expressions like "the Nth <noun>", the definite article is actually expressed by the ordinal numeral, so there is no article used by the noun. See this thread for more info.
Why isn't it 'doilea' ?
”doilea” is for masculine and neutral singular.
Couldn't this also be translated as "I am the second daughter."?
Not in this sentence. "I am the second daughter." is ”Eu sunt a doua fiică”.
But your guess is good. ”Fată” can mean ”daughter” in some specific context.
”Vecinul are o fată frumoasă” - ”The neighbour has a beautiful daughter (girl).”