Hello Deextr,
Thanks for that info and for choosing the Apple Support Communities. If we understand correctly, you are not seeing saved contacts in Caller ID on your iPhone after adding an eSIM. First, make sure your iPhone is up to date, as the updates contain stability fixes that can resolve issues like yours, as well as important security updates to keep your iPhone safe.
Update your iPhone or iPad - Apple Support
Manually update your carrier settings on your iPhone or iPad - Apple Support
Next, check if you have any call filters setup, as they could cause this behavior:
"Set up an app to filter and detect spam calls
1. Go to the App Store and download an app that detects and blocks spam phone calls. You can download and install multiple apps with this feature from different app developers.
2. Go to Settings > Phone.
3. Tap Call Blocking & Identification.
4. Under Allow These Apps To Block Calls And Provide Caller ID, turn the app on or off. You can also reorder the apps based on priority. Just tap Edit and then drag the apps in the order you want them.
Phone numbers that appear under Blocked Contacts are numbers that you manually block.

When you receive a call, your device checks the caller’s number and compares it to the list of phone numbers in your third-party spam apps. If there's a match, iOS displays the identifying label chosen by the app, for example Spam or Telemarketing. If the app determines that a phone number is spam, it may choose to block the phone call automatically. Incoming calls are never sent to third-party developers."
And that is from here: Detect and block spam phone calls - Apple Support If you are still having this issue after following those steps, please reach out to your carriers and see if there is a call filtering service on their end that could be causing this behavior.
Cheers!