You did a Restore? As a test, to see if there is something generally wrong with the iPod, set it up to do a small automatic sync. Select the iPod in iTunes 12, so that you see the iPod's "management" screen. In the iPod's sidebar, under Settings, click Music. To the right, check the checkbox for Sync Music. This enables automatic syncing for songs. NOTE: Automatic syncing replaces your iPod's existing content, so if there are songs currently on the iPod that are not in your iTunes library, don't do this. But if you already did a Restore, it's a non-issue since you've already erased the iPod.
Below the Sync Music setting, select the option to sync Selected playlists, artists, albums, and genres. Below that, under Playlists, select (checkmark) one playlists with a reasonable number of songs. Or select one artist under Artists. Click Apply.
Does that sync the songs in your selection to the iPod?
FYI - I have several iPods that are older than your 2nd gen nano, including a 3rd gen "classic" iPod from 2003 (which connects using FireWire). They all work fine with the iTunes 12.