Yes, you can use any Apple ID on any Apple product, regardless of who it is registered to. Simply sign in with your Apple ID (e.g., email@example.com) and purchase or download the app you wish from either the App Store or the Mac App Store.
Product registration helps Apple pull up your details quickly and easily should you (or your family) need warranty service. It's not at all required and the warranty of any Apple product is bound to the actual device, not the person. If you purchase an item from the Apple Store, you'll notice that your warranty already starts on your product purchase date. If you buy an item from a 3rd party retailer, you will have to bring in the original receipt to show date of purchase. We often lose or forget about receipts, so registering your product is often convenient for us as well as Apple.
If you purchase content using your own private Apple ID, that content is bound to your account. If your family ever wish to update an app, or redownload it, they will need access to your Apple ID account.
There is nothing that technically forbids you from using many Apple IDs on a single system, but often tracking who bought what gets to be a bit of a pain, especially when someone moves on. Just of note that it's often best to keep things simple and use a single Apple ID. It's important to remember that you can't transfer purchases from one Apple ID to another. You also can't ever close or delete an old Apple ID. When an Apple ID is created, it is quite permanent.