Add a device to the list

Clarification question - on the computer, did you sign into System Preferences/iCloud using the ID? That is what adds it to the device list. If you signed in with your dad's ID, it would be on the device list for that Apple ID, not your ID.

See Authorization vs. association - Apple Community for some background. A device is added to the list when you perform certain actions such as downloading past purchases. You can make use of your father's unprotected media by simply copying it to your library. For protected content you would need to authorize your computer to his ID. (I'm assuming you have your own computer at this point). As noted in the article attempting to download content from different IDs to a single computer may not always be possible. Associating this computer with a new ID is not something you want to do immediately before giving it away. Backup the whole computer if possible. You could also take a separate copy of the user profile on a second drive for added security. See What to do before you sell, give away, or trade in your Mac - Apple Support.

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I'm not sure if it is possible to have one device be a trusted device for more than one Apple ID. As mentioned in Authorization vs. association - Apple Community one workaround for being able to access purchases from more than one account on a single computer is to link the accounts together with Family Sharing, something you might be able to do before you wipe his computer. That said, as long as you already have all of the content downloaded and stored on an external drive you can use it as you see fit, authorizing your computer to his ID for protected content if required.

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From Two-factor authentication for Apple ID - Apple Support:

Trusted devices

A trusted device is an iPhone, iPad, or iPod touch with iOS 9 and later, or a Mac with OS X El Capitan and later that you've already signed in to using two-factor authentication. It’s a device we know is yours and that can be used to verify your identity by displaying a verification code from Apple when you sign in on a different device or browser.

From View and remove your associated devices in iTunes - Apple Support:

More about associated devices

Your iPhone, iPad, iPod touch, Mac, PC, or Android phone automatically associate with your Apple ID and iTunes when you:

• Sign in to Apple Music with your Apple ID.
• Download music to your iPhone, iPad, or iPod touch in the Music app to listen to when you're offline.
• Turn on iCloud Music Library.
• Download past purchases that you made in the iTunes Store, App Store, or iBooks Store.
• Turn on Automatic Downloads for iTunes Store, App Store, or iBooks Store purchases.
• Turn on Family Sharing.
• Subscribe to iTunes Match.

When you try to associate another Apple ID with a device that was already associated, you might see a message that says, "This device is associated with another Apple ID. You cannot associate this device with another Apple ID for [number] days." Depending on when you associated your device with your Apple ID, you might need to wait up to 90 days to associate the device with another Apple ID.

So you add a device to the trusted devices list by signing into your Apple ID on the device when two-factor authentication is enabled. You add a device to the associated devices list by using certain features of the iTunes Store.

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If you went to System Preferences/iCloud, why didn't you have an option to sign in? Was another ID signed in?

I didn't understand that you wanted to log into your dad's ID everywhere.