How can 3 different Apple Music users share the same Apple TV? (family or not family plan)
You can set up multiple Apple ID's on the Apple TV and have each user switch every time they use it.

You may opt to create a separate/new account that you leave signed in, so that somebody doesn't accidentally use yours (requiring you to sign in and out of your account every time).


  How to add multiple accounts to Apple TV
  
  Before you can use multiple accounts, you need to log into any and all
  additional accounts you want available.
  
  Launch Settings from your Home screen. Click on Accounts. Click on
  iTunes and App Store. Click on Add new Apple ID. Enter the additional
  Apple ID username. Click on Continue. Enter the additional Apple ID
  password. Click on Sign In.
  
  How to switch between multiple accounts on Apple TV
  
  While you have multiple accounts you can only use one account at a
  time. That means switching!
  
  Launch Settings from your Home screen. Click on Accounts. Click on
  iTunes and App Store. Click on the account you want to use.
  
  You'll now be able to download or re-download any apps or content
  available on that account.
  
  Family Sharing
  
  Multiple accounts work with Family Sharing, so kids can switch to
  their accounts and "ask to buy" just like on iPhone or iPad.