Duolingo's "early years" were fraught with an exaggerated emphasis of grammar, in the sense of "well, it could be that". And grammatically, "son" may be either "his" or "her". The reality is that "il cherche son chapeau" would be understood by a francophone as his hat. Otherwise, context would influence it. If there is a chance for confusion and the speaker wants to be very clear that he is looking for her hat, one would say "il cherche son chapeau à elle".
Hi, Margita. And just to add another piece of information about "type what you hear" exercises, Duo does not translate your answer and display it to you. The exercise is based on the sentence "il cherche son chapeau". So even if you write "ils cherchent son chapeau" (which sounds identical), it will still show you "il cherche son chapeau" (he is lookinig for his hat) as the intended answer. It is not a mistake--it's just the way Duo works.
"chercher", like "demander" is directly transitive, while "to look/search for", like "to ask for" need a preposition.
Verbs have to be learned as they come, with their own constructions, because prepositions are the less directly translatable words from one language to the other.
Besides, "un chapeau" is masculine, so its possessive adjective has to agree with the noun it modifies: son chapeau