Back up
If I backup my IPhone on a Mac and someone else backup their IPhone with a different ICloud ID, would the first backup be deleted???
iPhone 8 Plus, iOS 12
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If you are backing up in iTunes on a Mac, then your iCloud ID has nothing to do with making that backup. iTunes backups are named based on the device's name, so you can backup as many uniquely named iOS devices as you wish.
If you are also sync'ing iTunes content, then yes, the AppleID used with iTunes does matter. But you can simply each have your own MacOS account on the computer, and login and use your own Mac user account to backup your own devices. User accounts are separate, so they can each use that person's own AppleID.
I would recommend that latter method when mutliple people are using one Mac - make sure everyone has their own MacOS account on the computer and everyone uses their own account thereafter for everything they do on that computer.