Hi, my native language is American English. More often you will hear the definite article before ordinal numbers, but the use here is correct.
Sometimes there is no article (e.g. "The winner came in first place."). Sometimes you use the definite article (e.g. "The first thing I did was to open the door."). And sometimes you do use the indefinite article, as seen here.
In this example, the speaker says, "Now we have a third daughter." The word "now" helps signal that the circumstances have changed from the past. Presumably the family used to have only two daughters, but now it has three. The indefinite article is used to emphasize this change in circumstance.
One might say, "I bought a second car." It's understood that before, he only owned one car. One might also use the possessive instead. "On vacation in Rome, I met my third husband." It's understood that the speaker had two prior husbands. She may be on her fifth marriage, but she is currently talking about husband number three.
I hope that helps. Let me know if I can further clarify anything I wrote.
As a translation for now, I do not believe there is a difference. http://forum.wordreference.com/showthread.php?t=9473
https://www.collinsdictionary.com/dictionary/italian-english/figlio figlio - son (senza distinzione di sesso [without distinguishing the gender]) child. "Figlia" can only refer to a daughter.