Sorry, I mixed that up, Preferences is correct, Settings is the term on the iPhone 🙂
Just to make sure, the Apple ID is the same for each device, but the Mac has a mail address selected under send/receive at and the phone number is selected on the iPhone under Settings/Messages/Send & Receive at.
What happens if you use another mail address on the Mac?
I just checked iMessage on my Mac, I recognized that my conversations only remained on the Mac if I responded form my MacBookPro. Conversations I did write on my iPhone were not mirrored to the account on the Mac. For example, from one of my contacts, I still have one message from 2 years ago, just because this one response had been written on the Mac, not on the phone. the rest had been held on the phone and thus is not present on the Mac.
Also, I do not have continuity, since I'm still on OSX 10.9.5
Can you send an iMessage from your Mac to your iPhone and the other way? If yes, do they show up?