Hey where5334,
We'd be happy to help out. What you're noticing wouldn't be expected. When you have someone set up as an emergency contact, it should still show their name listed for them in their contact card when they reach out to you or vice versa. Perhaps your contacts aren't syncing as expected at the moment?
If you use iCloud to manage your contacts, make sure you still have Contacts enabled for your account by going to Settings > Contacts > Accounts > iCloud > Toggle on Contacts.
If you're using a different account for contacts than iCloud, then repeat the above steps, but tap on the affected account instead of iCloud.
For more help, check out: If your iCloud Contacts, Calendars, or Reminders won’t sync - Apple Support
We hope this helps out.
Take care!