Grammatically, yes. But logically it would be understood by francophones as "his beer". "Sa bière" can technically be "his beer" or "her beer", but in conversation it will be understood as referring to the subject of the sentence. A few examples:
- Il est tombé de son cheval = he fell off his horse.
- Il est tombé de son cheval à elle = he fell off her horse.
- Elle est tombée de son cheval = she fell off her horse.
- Elle est tombée de son cheval à lui = she fell off his horse.
finit and termine are interchangeable here (and most of the time elsewhere)
son, sa, ses = his
son, sa, ses = her
son, sa, ses = its
- son + masculine singular noun = son vin
- son + feminine singular noun starting with a vowel sound = son eau
- sa + feminine singular noun starting with a consonant = sa bière
- ses + any plural noun = ses vins, ses eaux, ses bières
I know that some sentences allow for "his" or "her" but not this one (I removed "her").
"Sa bière" can mean "his, her or its beer", so it does not give you any information on the owner's gender.
In theory, then, "the man finishes her beer" is possible, but in reality, the French would add something if the beer were not his: "l'homme finit sa bière à elle".
Conventionally, the object belongs to the subject.