Hello Panagioti Stouras,
Thanks for reaching out to the Apple Support Communities. I understand that you’re not seeing any new contacts that you’ve added when you check in your Contacts app, and these same contacts are showing as only their phone number in Messages on your MacBook Pro. First, you’ll want to get the contacts appearing in Contacts. Once they’re showing up there, you should see a change in Messages.
If you created the contact on your iPhone, you’ll want to be sure that you have Contacts turned on in iCloud, and that you’re signed into iCloud on your MacBook Pro. Here’s how to check if you’re signed into iCloud on your MacBook Pro: Set up iCloud on your Mac.
If you have iCloud signed in on your devices, and Contacts turned on, the next step is to check and see if the new contacts are appearing on the iCloud.com website. To check, sign in > select Contacts and see if they are present. If they are there, but not showing up in Contacts on your MacBook Pro, go to Contacts > Preferences > Accounts and see that you have the account enabled.
If you need more help, message back with the results.
Enjoy the day!