I guess it is because native English speakers would not commonly use the term "female friend".
(I believe Duolingo should differentiate between
1- "understandable, kind of literal translations",
2- "entirely correct, often used versions", and
3- "really wrong ones".
Then, it would be clearer whether we got the "German sentence" wrong or were just incapable of choosing the best possible "English sentence" for expressing the "idea".)
Eure specifies that instead of talking to one person, you're talking to a group of people. Which makes it weird, because you're talking about the girlfriend of a group of people. So we can assume that in this context, 'Freundin' just means 'female friend'. See the other comments for more info on that particular point.
Oh! I see what you're asking. So what you're getting confused by is the fact that you CAN use 'ihr-' as the 2nd person possessive adjective, but only in formal contexts. Think Sie vs du. In formal context, there's no difference between 2ndperson singular and plural, both are ihr-. (In this case, since 'Freundin' is female, it would be 'Ihre' - 'Ihre Freundin schreibt'.) But when talking informally, there IS a difference- 2p singular is 'dein-', 2p plural is 'eur-'.
So no, the difference between Ihr Eur isn't nominative vs accusative- it's formality. EDIT: check this link out for more info: http://www.germanlanguageguide.com/german/grammar/possessive-adjective.asp