Each "person" has a specific possessive adjective. Possessive adjectives agree with the thing owned, not with the owner.
- je => mon (masculine object), ma (feminine object), mes (plural object)
- tu => ton, ta, tes
- il/elle/on => son, sa ses
- nous => notre, nos
- vous (polite and plural) => votre, vos
- ils/elles => leur, leurs
Son/sa/ses is probably more accurately translated as "their", since it conveys no detail of the owner's gender.
"Their dog" would be fine in english, and determining gender would require context.
It is sometimes misleading when we look at the given translation and say it is "THE translation". As ZuMako has said, "son" can mean either "his" or "her". Both are correct (with no weird conjuring of explanation) and both are accepted. It is only when you have a sentence like "il a donné à manger à son chien". (He gave his dog something to eat) that the hearer will understand "son chien" as "his dog". Another example I like to use:
- Il est tombé de son cheval = He fell from his horse
- Il est tombé de son cheval à elle = He fell from her horse
- Elle est tombé de son cheval = She fell from her horse
- Elle est tombé de son cheval à lui = She fell from his forse.