The problem is, what would you expect him to do? Now this becomes an extrabiblical expectation.
Go to him and say "I want you to come to our church on Sunday, stand at the front of church and repeat this written statement where you swear to treat this woman as a wife" - and he'll say no, obviously, he doesn't want any part of that nonsense (from his perspective). That's an extreme example - but what level of "verbal claim" would you consider sufficient? Who are you to decide this is sufficient? Where did you find that in scripture? And why is it your business anyway?
But grab his woman's breast and he'll punch you in the face, because she's his woman.
That is such a ridiculous strawman response! I expect better of you. Your comments are usually well reasoned.
The OP asked for our thoughts and advice. I gave my opinion, clearly stating "I think" and "I would like".
That is totally different than "God requires" or "This saith the LORD".
I did not say that they were "fornicating", or that they were not "one flesh".
I said nothing about any formal religious ceremony or sacred vows. You made up all that up.
I clearly stated my basic social expectation, saying that he "should normally refer to her as his wife".
This is a very reasonable social expectation of a married man.
He claims that the woman belongs to him, and that this union is permanent..
How does he introduce her to a new acquaintance, to coworkers, friends, neighbors, or family? It is more accurate for him to introduce her as his wife rather than his girlfriend.
If you want to play semantic games about the fact that wife=woman in Greek and Hebrew, then fine. It is also perfectly accurate for him to say "she is my woman" instead of "she is my wife". That sounds a bit crude in our culture, but is also Biblically correct. It clearly defines the relationship.
The phrase "she is my girlfriend" is vague and suggests a temporary arrangement of convenience rather than lifelong union.
Why do you apparently think it is acceptable for him to use the confusing terminology of "girlfriend" instead of the clear and accurate terminology of "wife or woman"?
I'd also like to add another piece of advice. I think the woman should take his last name (whether officially/legally, or just informally). Seeing that she belongs to him, she should be called by his name.
A believer in Christ who refers to Jesus as merely "My Heavenly Friend" rather than "my Lord and my Savior" is badly mistaken.