No, you can’t change the ownership of a backup. As a workaround you can sync data to iCloud (which is completely different than backing up to iCloud. Here’s an explanation:
iCloud+ has 2 independent functions. iCloud backups, and iCloud sync. It’s confusing because they both have the same name prefix. But they have no connection with each other. Go to Settings/[your name]/iCloud. You will see a bunch of switches (with iOS 16 also tap Show All). When you turn on a switch that data type will sync to iCloud. If you do this on multiple devices that share an Apple ID the selected data will sync to all of those devices.
You can also turn on iCloud Backups. This will back up your phone every night if the phone is plugged in, connected to Wi-Fi and locked.
But note that anything that you sync by the first method will be excluded from the iCloud backups.
Here’s how to use the workaround:
- Log in with the Apple ID that has the desired data if not already
- Go to Settings/[Apple ID name]/iCloud
- Turn on the switches for data you want to transfer to a new Apple ID
- Wait for it to sync; leave the phone plugged in and connected to iCloud
- Log out of iCloud, and when asked choose to keep the data for the synced items on the phone.
- Log in with the new Apple ID.