The reason it is written as "fete" here is because that is the dative form of "fată." Since girl is an indirect object here, it must be in the dative case. If you look in dexonline.ro/ and click on the declinations tab when viewing a noun definition, you will see all of declensions for the noun by case, gender, and number.
It doesn't quite mean the same thing - "some girl" here has a negative implication about the girl, that she isn't approved of by the speaker or at least that him spending his time making soup for her isn't approved of! Stick with 'a girl' unless you are emphasising the smallness of his enterprise, perhaps in contrast to someone else's: "He makes soup for one girl, she makes soup for the whole street"
I just asked my teacher this question. She says that both formulations mean the same thing in this case, that is, "pentru o fată" or "unei fete." She did admit however that in this case, she slightly prefers the second, dative formulation when she speaks. As an example, she would often say, "Vrei să-ți fac un ceai?" but would use pentru in examples like this, "Am făcut aceste sarmele fără carne pentru mine, pentru că eu mănânc cu carne." At the end of it all however, she maintains that both ways are interchangeable without a change in meaning.
I find this hard too but I think it works like this:
In general, to form the genitive/dative form of a feminine noun, take the plural form and add an "i"
una: "one" feminine singular (rather than "un" or "unu" because "girl", by which I really mean "fată", is feminine). Actually, it is also articulated (note the "a" without the accent) because it is a pronoun rather than an adjective, so perhaps it should be rendered as "the one" in English.
Une: "one" in the feminine "plural" form that is "the ones" (I know that sounds weird but see comments about definite/indefinite articles below)
Unei: "of one" (genitive) or "to one" (dative) (or perhaps "of the ones" or "to the ones"---see above)
That is, "unei" is the pluralised form of "una" with an "i" appended to the end, in keeping with the rule stated above.
Coming back to the weird concept of a plural form for one, consider the definite and indefinite articles in English.
The definite article in English for both singular and plural forms is "the" (eg "the book" versus "the books")
The indefinite article in English is "a" for singular and "some" for plural (eg "a book" versus "some books").
So "unei" could also be thought of as "of some" or "to some" (for genitive and dative, respectively). In fact, in the correct context, it is sometimes translated as "some", which makes sense if it is thought of as the phrase "of the ones", particularly as it is used for countable nouns (the Romanian word for "some" in the nominative/accusative, including uncountable nouns, is "niște").
So this might be thought of as "He makes soup for some girl" with "some" being used to denote indefinite rather than definite (ie "for a girl (not otherwise specified)", as opposed to "for some particular girl"). As "some" is being used as an indefinite article in its genitive/dative form, and, as we do not have an equivalent declension in English, this becomes "He makes soup for a girl" (noting that the indefinite singular genitive/dative form of "fată" has the same spelling as its plural form "fete" and it is used in this grammatical construction because it is an indirect object---flipping hard, isn't it LOL). Actually, this is a better attempt to explain why I asked the question in the post I made earlier in this thread.
For completeness, I think the masculine (and neuter) singular equivalent of "unei" is ""unui" so the "unui baiăt" would mean "to a boy" in the sense of "to some boy" or "of some boy" (for the genitive and dative, respectively). So "El face supă unui baiăt" would mean "He makes soup for a boy" (as if to add to the confusion, the spelling of the indefinite genitive/dative of the masculine noun "baiăt" is the same as that of the nominative/accusative). The plural genitive/dative for all masculine, feminine and neuter cases is "unor" so "El face supă unor fete/baieți" would mean "He makes soup for some girls/boys". To emphasise the change note that it is the "unei" that has changed to "unor" (noting that the indefinite genitive/dative form of "fată" is "fete" whose spelling makes it indistinguishable from the indefinite nominative/accusative plural form).
Disclaimer: Sorry, I have rambled a bit. Nevertheless, this is my attempt to understand the translation offered for this sentence. Hopefully, someone who knows Romanian grammar better than me might come along and refine it, if necessary. In the meanwhile, this seems to work as a pragmatic approach to understanding this specimen sentence.
Thank you both DavidONeil4 and cheerfulcharlie for your responses, very helpful and detailed! It seems more similar to the German indirect object than I realised, which helps. The use of the same ending for plural and dat/gen is not so helpful, but I'm sure it'll all make sense eventually haha!