Hummm,
You mention Continuity in your first post.
I recently came across a post where this might have been set whilst the two Messages app had not been set up to link in the first place:-
Re: Re: Messages icon on Mac next to doc
Lets check how you set things up. (in the order the features arrived).
1) Enable the iPhone for Messages (allows it to send blue iMessages to Apple IDs and to iPhone Numbers that are registered).
2) Enable the SMS option is case you are sending to non iPhone.
3) Enable the Mac version by adding the Apple ID. (it can then send blue iMessages to iPhone Numbers and Apple IDs that are registered0
Option - test send and iMessages in each direction
4) Add the Apple ID to the iPhone. This allows iMessages sent to either ID/Number to appear on all the devices listing both items.
Adding it to the iPhone triggers a pop up on the Mac version that you accept to show the iPhone Number there.
5) Enable the Text Forwarding to get the iPhone to send SMS item on to the phone.
The iPhone has to have WiFi on for this and have joined the LAN the Mac is on.
Enabling the Forwarding triggers the slide out pane with the Code Number to appear on the Mac which you have to enter back on the iPhone before closing the Forwarding option.
Resetting this to "force" a new Code to appear can take several goes before it "releases" the Mac from the previous set up.
6) Continuity/Handoff
Only certain Macs can do this and it also has to be on at least Yosemite
See this Doc OS X Yosemite: Pick up where you left off with Handoff
My Mid 2011 iMac i5 21" cannot do this and I do tend to forget about it.
As I have stated it is probably the last "feature" to be added to the iMessages/SMS linking but as I suggest in the other thread I have not heard of it being set up without going through the other settings but I do think it s possible.
If it still is not obvious where the issues is I would consider resetting everything from scratch.
Sign out of the Apple ID on the iPhone and then set it to Airplane Mode.
Sign Out on the Mac and close the app.
After a couple of Minutes go back to the Messages setting on the iPhone and wait from the number to Verify.
Add back the Apple ID on the Mac version.
Once the Number has verified test and iMessages specifically addressed to the Apple ID.
Return the iMessages from the Mac.
If that works add back the Apple ID on the iPhone.
Accept the Pop ups on the Mac about what the iPhone Is using.
Enable Text Forwarding (SMS).
Enable Handoff where required.

9:18 pm Tuesday; February 23, 2016
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad