Can I add my old iPod to my Apple ID account?

An iPod does not need to be added to your account.

There is no native way to get music from an iPod back into an iTunes Library. The music on it should still be in the iTunes Library that it was synced with originally. Do you still have access to that, or a backup of your music?

If not, you will need a third-party programme to get the music off the iPod and into a usable form on your computer. Fellow user turungtest2 has written an article on this, which you can find at Recover your iTunes library from your iPo… - Apple Community. Some of the software he recommends is free, some is not.

The iPod you are using (6th gen, MC297) is not able to be synced to an Apple ID or placed under an iCloud activation lock, I even have difficulty trying to get music off old iPods, you need to get a 3rd party program to do so.

The iPod touches are the only iPods that you can associate with Apple IDs. An iPod touch runs iOS, and is very similar to an iPhone, the radio hardware.

All of the other iPods were, more or less, dedicated media players.