The usage of "his woman" would be routed from translation of Bible from Hebrew. In Hebrew, there is only one word for (Ish) man, and husband, or (Isha) woman, and wife.

English translations translated freely taking preference for poetic sound.

So as long as the bible is part of English speaking culture the usage of woman as wife will be common.

Something I find interesting about Persian culture. In Persian the word Zan means both wife and woman, but there is another more formal word Khanom. Which is always to be used when speaking or referring to someone older than you. I wouldn't be surprised if this is due to the etymology of the two word being from different languages.