Welcome to Apple Support Communities 0358,
Thank you for posting! From our understanding, in the Messages app, your contacts are showing only the number and not the saved name, is that right? Does this continue if you close the app out completely and relaunch it? How about if you restart your iPhone?
Do you sync your contacts with iCloud? If so, you can try toggling contact off in Settings > Your Name > iCloud and then turning it back on to force it to sync again.
Check out the following resources:
Close an app on your iPhone or iPod touch
Restart your iPhone
If your iCloud Contacts, Calendars, or Reminders won’t sync
Hope that helps!