Translation:My sister does not like chicken, however, she cooks it very well.
But своя would be a possessive. Possessives aren't really relevant here. An example from English: She is eating her food. She is eating her. Not only do these sentences differ slightly in meaning, the different versions of "her" fulfill very different grammatical functions. There is no word like "food" on which the possessive could be used. So the only two options are putting the subject in there a second time, but then you would need она, or having an object in there. And the object is simply "her" as in the second sentence with the chicken being the "she".
By the way, here is the explanation:
Functionally, they're the same; my guess is the course writers didn't code the phrase "even though" into the answer. They have some slight differences in shades of meaning. Although emphasizes the contrast and focuses attention to the second item contrasted, and can only be used at the start of a sentence: "Although injured, I ran a mile in 8 minutes." Even though implies a sense of spite or incredulity: "I ate the peanuts, even though my wife is allergic." An extreme example, to be sure.
"Но" is just "but". For example, it would be used to say something like "I like ice cream, but I'm lactose intolerant." "Зато" is also "but", only in a more "on the other hand" or "despite that" way. For instance "I don't like avocado, but I know how to cook with it."
Whereas но presents merely a foil or reason against the preceding sentence, зато is more set against the assumption. For example, it would be assumed that because my sister doesn't like chicken, she would have nothing to do with it. But (зато), even though she doesn't like it, she cooks it very well. As opposed to my sister knowing how to cook well, but (но) not knowing how to cook chicken because she does not like it.
[ но за-то́ ] is a [ за-то́ ] synonym. They may be adjacent in the same clause. ‧ ru.wiktionary.org/wiki/зато ‧ context.reverso.net/translation/russian-english/Но+зато+здесь ‧ context.reverso.net/translation/russian-english/но+зато ‧
но ‧ but
за-то́ ‧ but on the other hand
зато (negative, зато positive) ‧ www.duolingo.com/skill/ru/Conjunctions/tips-and-notes ‧
но за-то́ ‧ [ but ‧ instead ‧ ] ‧ ‧ context.reverso.net/translation/russian-english/но+зато ‧ ‧
ЗАТО НО difference ‧ www.youtube.com/watch?v=62Rxx3qV8Hc ‧
I think that this is a fairly common problem on several courses, particularly with the longer sentences. The missing words are there, but they are hidden by a frame at the bottom of the screen. Best you can do, until they fix it, is to switch to the "use keyboard" option and type in the answer manually.
Some people might naturally say this, but it's poor grammar in English. "Good" is an adjective, and modifies nouns (good cook). "Well" is an adverb and modifies verbs (cooks well). Plus it wouldn't be accepted for the use of "but" instead of "however" for reasons in comments above.
This sentence appears very similar to another in this lesson, "She does not like soup, but cooks it very well." In the other sentence, "but" is но, but here, it's зато. As far as I can tell, "but/however" plays an identical role in both sentences, so why the difference in Russian word choice?
Thanks! I think teaching both makes sense, though I left out in my previous post that they marked me wrong when I answered with но rather than зато in this particular sentence. I was confused by the inconsistency, which suggests a distinction between the two words that isn't clear from the lesson.
Actually, a bit of Googling for "difference between но and зато" found me this: https://hinative.com/en-US/questions/3821197 which somewhat covers the same ground as a comment by TheRaginPagan up the page a ways. So there is a difference, but if both accepted in other similar sentences, it's strange that that both are not accepted here.
Thanks! Agreed, the inconsistency is confusing. Based on the previous response and your link, it looks like the use of зато is correct in this case, so I'm not sure why the other sentence in this lesson used но. Hopefully someone at Duolingo will take another look and correct the translations.