The problem of course is the presence of the pronouns which do not give any differentiation between people. Duolingo wants the idea "This man is buying the other man something to eat and drink". Have you reported this ambiguity?
"He is buying .." translated with "il lui offre ..." is exactly the inverse of the sentence correct? I think duolingos answer is wrong or at least very far fetched...
I missed this too, so I looked it up. Here's what I found: "The posessive adjective must be repeated before each noun." So Duo is right and we're wrong.
"Schaum's Outline of French Grammar: Fifth Edition" (Crocker, M; 2009, p. 55)