I was wondering whether it had that connotation like in French. I would guess if it's not accepted that means the language doesn't imply that. Google translate gives "fiica mea" for "my daughter", so whereas in French there isn't a word to differentiate between girl and daughter, but only the possessive adjective, it seems like in Romanian like in English there is a different word so we're expected to translate "fata mea" directly to "my girl".
Why not "A opta fata.."? We are speaking about a definite girl - the eighth one - so why use the indefinite form 'fată'?
Because the definite article is already covered by the numeral.
This is generally true for cases in which you switch the noun-adjective order - the noun loses the definite article which is attached to the adjective:
The beautiful girl is there. -- Fata frumoasă e acolo. / Frumoasa fată e acolo.
In the sentence at hand, you have to do this because you can't place the numeral after the noun.
It adds emphasis on the adjective. These noun-adj inversions are very common in literature - they are perceived as artistic. Plus, from a technical point of view, it's a great way to control rhythm or rhyme in a poem:
A fost odată ca-n povești,
A fost ca niciodată,
Din rude mari împărătești,
O preafrumoasă fată.
(Mihai Eminescu - Luceafărul)
You can see a switch, but with the indefinite article; also "preafrumoasă" means "very beautiful".
Disclaimer: I'm just a native speaker with an active interest, I have no serious formal training in linguistics. This exact topic seems very complex to me; besides what's described above there are other situations in which the noun-adj order is changed and I probably couldn't exhaustively cover them all.