Your shuffle is not like your iOS devices. The shuffle itself is not involved (in any way) in purchasing and downloading from the iTunes Store. Your computer running iTunes is doing that part, and it does not care if you have a shuffle or not. Therefore, for your gift card, the IMPORTANT step is to make sure you are signed in to the iTunes Store using your Apple ID, on your computer.
Unlike your shuffle, your iOS devices (iPhone and iPod touch) ARE computers. They are used to purchase and download from the iTunes Store. That is why they need to be signed in ("registered") to your Apple ID.
Once you have added all of the songs to your iTunes library, whether the songs came from the iTunes Store, music CDs, or other source, THEN you sync your songs to the shuffle as desired. It does not matter if your shuffle is registered with Apple or not. That is just a serial number tracking procedure that has absolutely nothing to do with using the shuffle (or with using a gift card to buy songs from the iTunes Store).
It is important that you use the same Apple ID with the iTunes Store on the computer, that you use with your iOS devices. This is because you can reload previous purchases at no cost. So, if you purchased songs using that Apple ID on your iPhone or iPod touch, you can redownload them using the iTunes Store on your computer. Go to the iTunes Store Home screen, and click on Purchased under QUICK LINKS. You need to be signed in using that same Apple ID. On the Purchased screen, you can re-download previous purchases made using that Apple ID.