As far as I understand this is literally "He is washing hands", with it being implied but not stated that it is his own hands that he is washing. In English the sentence would be a bit weird, is it normal in Ukrainian to miss it out the "his" like this?
Yes, it is completely natural for Ukrainian. Even opposite: "він миє свої руки" would look natural just in case there's some context where some other possibilities where mentioned.
Just like Polish "On myje ręce" - "He washes/is washing (his) hands" :)
I know in Russian that anything to do with your body, you won't use a pronoun or possessive or things like that, so if you were to directly translate «он моет руки», it would look really strange. I'm certain it's the same in Ukrainian