It is a borrowed Word from English (or French, not sure). Duolingo is just using the Translation 'Coat' to confuse us.
Even worse is 'Makarna', that Duolingo translates to the italian Word 'Pasta' instead of the english Word 'Noodle/s'. Why do this when 'Pasta' is also a turkish Word with a different meaning?
yes, the problem is that the accusative -(y)I and possessive endings -(s)I are almost the same. i believe the hints should be improved, that hint was probably added for another sentence.
ceketleri [accusative] gördüm. - I saw the coats.
kadınların ceketleri [possessive] burada. - The women's coats are here.
kadınların ceketlerini gördüm. [possessive + accusative] - I saw the women's coats.
also note that possessive -(s)I becomes -(s)In when it's used with any case.
sure you can, "the women's coat" would be "kadınların ceketi"
plurality is only ambiguous if you are talking about an indefinite object in a sentence . and even then, the plural in English is not a real plural, they use plurals to talk about a general thing and we use singulars
I'm looking at an example in my book. It says "cocuklarin bahceleri" means either "(the) garden of the children" or "(the) gardens of the children". hence "kadinlarin ceketleri" means "(the) coat of the women" or "(the) coats of the women". It says it's because you can't duplicate the "lar" ending. It also says you can put a singular possessive ending on the possessed object like you did, but it doesn't say that's required, and I wonder if it's informal.
the comment at the top of the page you linked doesn't talk about possession.
(i can't type the special characters on this computer.)
OK I see your point, and that's true. Thinking of it, I believe we usually do this for things that they actually share, like "onların kedileri" could be "their cat", or "onların kızları" could be "their daughter" - you are right. But for singular forms, we can well use the singular forms too, so saying "onların kızı" is perfectly fine.
and here, it is really weird that they share a coat, so I never taught of it.