iPhone 7 iOS 10.3.3 Contact missing after logging into iMessage on Mac
I accidentially figured this out myself poking around with the same issue. My contacts are stored on my company's servers. So I changed the default contacts to my iPad and this fixed the issue. I then put my contact default back to my company's server and it still works.

Go to Settings > Contacts 
Look at the Default Account value and note what it is. You will need this for the last steo.
Change the Default Account (hopefully you have more than one) to iPad
- Or, just switch it to someting other than what it is.
Check your messages and see if the names show up instead of phone numbers
Go back to Contacts and put the default back to whatever it was orginally.

This worked for me, hope it works for others. 
Good Luck