Those aren't the vowel sounds that I hear. I hear "bonyeh", but there is a lot of variation in Irish accents, so you will hear different speakers use slight different pronunciations of the same word.
You can hear examples of 3 speakers from 3 different parts of Ireland pronounce bainne here: http://www.teanglann.ie/en/fuaim/bainne
Hover over aici again. It doesn't hint that aici means "she has", it hints that tá ... aici means "she has. It's the same when you look at the hint for tá.
You need both the tá and the ag X parts if you want to say "has" or "have".
Tá úll ag Pól - "Pól has an apple"
Tá arán aige - "He has bread"
Tá bainne aici - "She has milk"
Tá hata agat - "You have a hat"
Tá bainne agam (ag mé) - "I have milk"
Tá bainne agat (ag tú) - "you have milk"
Tá bainne aige (ag é) - "he has milk"
Tá bainne aici (ag í) - "she has milk"
Tá bainne againn (ag sinn) - "we have milk"
Tá bainne agaibh (ag sibh) - "you/ye/you'all have milk"
Tá bainne acu (ag iad) - "they have milk"
NOTE!!! the elements in parentheses () are not real - they are just included here to show where the combined forms came from. You never write or say "ag mé", etc.
There is no "the" in Tá bainne aici. It's just "She has milk" or "She has got milk".
The string matching algorithm that Duolingo uses decided that your answer was closest to "She has got milk" (it has the same number of letters as "she has the milk"), so it suggested that as the "correct" answer, but the default answer for this exercise is actually "She has milk".