Find My problem
I did the recent update. I am the main organizer of my family account but I do not show up correctly in the new Find My app. I show up as my son, who now shows up 2x in the Find My app
iPhone 7 Plus, iOS 13
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Apple IDs aren't meant to be shared. Everyone should have their own Apple ID to prevent things like you are seeing.
If 13 years old or older.
If under 13 years old, you will need to set up Family Sharing to create an ID.
Family Sharing and Apple ID for your child
When changing to the new ID, there may be some data loss, but you can minimize it. If syncing photos with iCloud, ensure Optimization is turned off in Settings/Your Name/iCloud/Photos and/or in Photos/Preferences/iCloud. Then make sure full resolution photos are on the device/computer. When signing out of iCloud, if asked whether you want to retain data, choose yes. Messages conversations will be lost if they are addressed to the ID. See the article below for other places that the old ID should be removed. See Winston Churchill's response in the below discussion. Also set up Family Sharing to keep access to your apps/purchases.
How to transfer all files from one iCloud account to another
Apple ID email address or password - What to do after you change