iPhone – Set Different Screen Time App Limits on Different Devices

I don't believe there are any software differences at all between an iPad mini and an iPad 2 or iPad Air. Both iPad minis use the exact same display resolution as their larger counterparts (1024x768 non-retina, 2048x1536 retina), just on a physically smaller screen. The whole purpose of this is to ensure that all existing software works identically on both device sizes.

In fact, I don't think it's even possible for an app to "know" which size device it's running on, since there's no technical way to differentiate them. Apple's own iOS Simulator doesn't even list the iPad mini as a separate device to simulate: the only options are iPad (meaning 1024x768) and iPad Retina (meaning 2048x1536).

It's all detailed in the Apple Support Article: View and remove associated devices in iTunes:

When you use certain iTunes Store features on your iPhone, iPad, iPod touch, or computer, your iOS device or computer is automatically associated with your Apple ID in iTunes account settings.

and

You don't need to associate your devices to buy content, download new purchases, or to play back content that you've saved to listen to offline.

To remove a device:

Under the iTunes in the Cloud section, click Manage Devices. Click Remove next to the device that you want to remove.

Note that only computers are Authorized and Deauthorized, and are limited to 5 per account. See this other article: Deauthorize your computer using iTunes

Finally, this Discussion gave a good answer on Authorized vs Associated:

A computer can be AUTHORIZED to play/use content purchsed from multiple iTunes Accounts. There is a limit to the number of computers that can be AUTHORIZED per account (5 total) but no limit as to how many accounts a computer can be AUTHORIZED to play. Once you have AUTHORIZED all the accounts AND downloaded all the purchased content from each account to a computer, this content can be also be played on any device ASSOCIATED with the computer. A device can only be ASSOCIATED to one computer but one computer can have many devices ASSOCIATED to it. As an example, you can use apps purchased on from multiple iTunes accounts on an iPad as long as the computer the iPad is ASSOCIATED with is AUTHORIZED to use/play the content for each of the accounts. Another example: if you also purchased music from all the iTunes accounts, you could listen to it on an iPad, iPhone and iPod as long as all three were ASSOCIATED with the computer where all the accounts were AUTHORIZED.