Good afternoon RobieDigital,
Welcome to the Apple Support Communities!
I understand the details that you are seeing in Contacts on your iPhone are not reflected on your Mac and is affecting what is viewed in Messages. Before troubleshooting this issue, make sure you have a backup of your iPhone and Mac.
Back up your Mac with Time Machine
Try restarting both devices initially to refresh them and reestablish a connection to the internet.
If you have not already done so, I would advise enabling iCloud for your contacts on the iPhone and on the Mac. This will sync the data between the two devices. If you do have iCloud enabled on both devices, toggle the iCloud Contacts setting off, restart your Mac, and then turn it on again to force a new sync.
For best results, only use the iCloud account to sync contact data. On your iPhone, go to to Settings >Passwords & Accounts to review the data that is stored in each of your accounts.
Check the Contacts settings on your iPhone. Open the Settings app and choose Contacts > Short Name. Try toggling off the Short Name (and Prefer Nicknames) options if you have them on.
Set up iCloud on your Mac
How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac
Please reply to this thread if the issue persists after trying the steps above, and I'll do my best to help you with this!
Take care.