Good evening Emma_NZ,
Welcome to the Apple Support Communities!
It sounds like you are having issues with birthdays and notifications in the Calendar app on your iPhone SE. Let's see if we can figure this out together.
It looks like you are using iCloud to sync your Calendar data.
Do you have a different iOS device or a Mac with iCloud Calendar enabled?
When you go to www.icloud.com and sign in with your Apple ID, are you seeing the same data online as you are seeing on your iPhone?
Is the issue happening with this one contact, or are you getting multiple notifications for birthdays for several people?
Have you updated your iPhone to iOS 14 yet? If you have not, let's get this issue resolved before updating the iOS.
You've done some good troubleshooting by turning off the default Birthdays calendar.
Before you attempt any more troubleshooting steps, make sure you have a backup of your iPhone and your Mac (if you have one).
How to back up your iPhone, iPad, and iPod touch
Back up your Mac with Time Machine
- As we begin troubleshooting, we'd suggest you turn off (remove the checkmark) from the "Birthdays and Anniversaries" calendar you created in iCloud - Set up multiple calendars on iPhone - iPhone User Guide. Keep the default Birthdays calendar (with the red gift) turned off. As you have set up the birthdays in a separate calendar, you will have duplicates if you also enable the Birthdays calendar which pulls information from the Contacts app.
- Write the information down that you have for "Stephanie" above and delete her contact card for now - Manage and delete contacts on your iPhone, iPad, or iPod touch
- Turn off Notifications for Calendar on your iPhone - Use notifications on your iPhone, iPad and iPod touch
- Check to see if you are storing contacts in any other accounts besides iCloud. Make sure iCloud is your default account. Follow the steps in the resources below carefully:
If you see duplicate contacts after setting up iCloud Contacts
If your iCloud Contacts, Calendars, or Reminders won’t sync
- Restart your iPhone.
- Enable Notifications for Calendar again on your iPhone.
Please reply to this thread to give us an update on your issue after trying the troubleshooting steps above. We'll do our best to help you with this!
Take care.