Q: How to merge iOS messages conversation feeds?

Unfortunately you can't merge them back into the same conversation.  To fix this, try going to Settings>Messages>Send & Receive, tap the ID, sign out, then sign back in.  You may both need to do this.

I had a strange problem with my son that started a different conversation.  It started with a text that I sent, and continued in the new discussion.  What finally stopped it for me was when I singed out and back in, then he went back to the original conversation and sent a text to me.  When I replied to that it stayed in the original (old) conversation.

Can't merge the conversations, but as far as getting devices and mac back syncing messaging properly, this article is very helpful.

iOS and OS X: Link your phone number and Apple ID for use with FaceTime and iMessage - Apple Support

Perhaps iMessage unintentionally started a new conversation to/from one of your email addresses (rather than the phone number). This has happened to me a few times.

On the two conversations, tap the recipient (in this case your wife), then tap the Info button. Perhaps one of the conversations highlights her phone number, while the other highlights an email address.

If this is the case, unfortunately there is no way to merge the conversations. The fix is just to manually check that you are replying to the correct conversation (phone number, likely), and let the other conversation slip into the abyss. Or just delete it.