It's not the accusative form, it's the possessive form.
The third-person possessive ending is the same as the accusative ending for words that end in a consonant, but different for words that end in a vowel, such as "kedi" (cat) -- "kedisi" is the possessive, "kediyi" the accusative.
The accusative of "their wine" would be "onların şarabını" with an -ı for possessive and an extra -ı for accusative (with buffer -n- between a possessive ending and a case ending).
If you say şarabı içerim it means "I think the wine", yes.
That uses the accusative case of şarap, şarabı, which happens to look the same as the third-person possessive form şarabı "his/her/their wine".
The accusative case is used when an object is definite, so "I drink (some) wine" would just be şarap içerim, while "I drink the wine" is şarabı içerim. I don't think it's helpful to say that the accusative ending "means" the, though.
"The wine is cold" would be Şarap soğuk, for example -- the English "the" is not translated there at all (because the wine is not an object).
And in onların şarab-ı, you have not the accusative ending but the possessive ending.
Similarly, for "my wine" it would be benim şarab-ım.
That can then be placed into the accusative case as well: onların şarab-ı-n-ı içmem, benim şarab-ım-ı içerim "I do not drink their wine; I drink my wine".