Turkish confusingly has two endings that look the same but mean different things.
You're thinking of the accusative ending, which is -i after consonants (subject to vowel harmony) and is used for definite direct objects, as in Portakalı yer. "He eats the orange."
But here we have the third-person singular possessive ending which is also -i after consonants (subject to vowel harmony) and is used to show that something belongs to "him, her, it" -- kızın portakalı "the girl's orange", or literally "of-the-girl her-orange", and kızın bir portakalı "an orange of the girl's" (literally "of-the-girl one her-orange").
Those two endings look differently after vowels - there, you can distinguish (for example) elmayı and elması -- the first is accusative, the second third-person singular possessive of elma.
But it's not really any worse than the English -s ending which can be plural, possessive, or third-person singular verb ending! (You can't tell from listening whether something that sounds like "roolz" comes from a sentence like There are many rules that apply here (plural), This rule's validity ends in March (possessive), or The king rules with an iron fist (verb ending).)
There are two endings which can look very similar: -(n)in is the genitive ending, and -(i)n is the possessive ending for "you".
After a consonant, when the -n- of -nin is not needed but the -i- of -in is, the two look the same, so kızın can mean either "the girl's; of the girl" or "your girl".
In this case, it's the genitive. Kızın ... yok means literally "the girl's ... does not exist" or better "the girl does not have a(n) ...".
The endings are distinct after a vowel, e.g. babanın "the father's" versus baban "your father".
It doesn't really work in English. "Any" is a strange word, and in a lot of sentences, it can't be used with singular, countable nouns. This is one of those sentences. "The girl doesn't have any water" would be okay, because water is uncountable. "The girl doesn't have any oranges" would also be fine, because "oranges" is plural. But the Turkish sentence only has the one orange, so we can't use "any" in the English one.
(I don't know if there's a good rule about "any." Generally speaking, if you have a "[somebody] doesn't have [something]"-type sentence, and the "something" is singular and countable, you won't be able to use "any." But "I don't have any idea" is an exception, and there may be other exceptions.)
"Kızın" is a possessive form, so in a lot of sentences, it will be "the girl's." (For instance, "kızın kedisi" would be "the girl's cat.") Duo auto-generates the hover tips, and if "kızın" often means "girl's," that's likely to be what goes in the hints. It doesn't work as a direct translation in this sentence (mostly because Turkish and English are so different in their treatment of "I have X, I don't have Y"- type sentences) -- but I do think that hint is useful, because it tells you that the sentence is going to deal with possession.
kızın can mean either “your daughter” or “the girl’s”.
In a possessive sentence, you need the possessive meaning, as kızın bir portakalı yok would translate word for word to “the girl’s one her orange does not exist = the girl’s orange does not exist = the girl does not have an orange.”
For the meaning “your daughter does not have an orange”, you would need the possessive form of kızın, i.e. kızının.