You should be able to add music to it from your iTunes Library.
If you have a more recent Mac, using OS Catalina or later, there is no iTunes. Instead, you use Finder to add music from your computer to the iPod.
You will not be able to access the store from the iPod, due to the age of its iOS version, which you will not be able to update.
You do not need to sign into your Apple ID to use iTunes (the programme). However, if you bought music from the iTunes Store, you will need to sign into the store in order to download those purchases to your iTunes Library and then copy those onto your iPod. I don't think that you can use Apple Music (the subscription service) on an iPod using iOS 6.1.6. Why are you unable to sign in (log in) to the store, what happens when you try?