Hi threekarat,
To help with seeing the names of contacts on your devices, verify the settings are correct for your contacts. On both your devices, go to Settings > Contacts > Accounts. Verify which account has contacts enabled and make sure it's the same between your iPhone and iPad. This should help sync the data from the server to your iPhone and iPad.
If the same issue continues, try restarting. This process can help refresh the settings and help resync the data. Here's how to complete this on both devices:
* iPhone with Face ID: Simultaneously press and hold the side button and either volume button until the sliders appear, then drag the Power Off slider.
Turn iPhone on or off - Apple Support
* Turn off iPad: Simultaneously press and hold the top button and either volume button, then drag the slider.
Turn iPad on or off - Apple Support
Test and report back with the results.
Take care.