Hi,
It sounds more like the settings about which ID or iPhone Number is being used to send iMessages.
On both devices there are "Start From" settings.
When you start a conversation this ID is used.
If you Contacts don't know that ID then their Messages app will identify you only by the ID or iPhone number used.
Lets say you originally set up iPhone and Mac to use the iPhone Number as the Start From ID then your Contacts are likely to have created and Address Card with that info on it so that you show up as "kewlpanda" for example.
If the settings have changed on one so that new Conversation Start From the Apple ID they see something like "kewlpanda(at)icloud.com" instead of your Name as they don't have that info on their Address Card for you.
Of course when they return an iMessage it will arrive as if it is a different conversation to you as well as to them because the "From" in the original to them is different.
8:16 pm Friday; January 8, 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