So, it sounds like the iPod DOES appear in iTunes (on the Windows 10 PC). That's good, because I don't think there is anything wrong.
If the iPod is currently being synced using the Windows 7 PC, the iTunes library on that computer is the only one that will sync to that iPod (until you change the association to a different iTunes library). Therefore, when you click Sync in iTunes on the Windows 10 PC, nothing is synced because that iTunes library is a different iTunes library that is currently not set up to sync that iPod. By definition, "syncing" (as in synchronizing) means iTunes is keeping the iPod's content the same as the associated iTunes library (or a selected portion of that library); iTunes cannot sync two different libraries to one iPod.
If you want to change the association to the iTunes library on the Windows 10 PC, select the iPod in iTunes so that you see its settings screen in iTunes window. Along left side of window (in the sidebar), under Settings, click Music. To the right, you see the iPod's Music settings screen, where you tell THIS iTunes library how to sync songs to the iPod.
NOTE: If you proceed, iTunes replaces the iPod's current content (from the other iTunes library) with content from this iTunes library. Any songs on iPod that are not in this iTunes library will be lost. Syncing is typically in one direction only, from iTunes library to iPod.
At the top, check the box for Sync Music. You should get a warning, because the iPod is currently set to sync with a different iTunes library. Below that, if your iTunes library (on this computer) can fit on the iPod completely, you can choose the option to sync Entire music library. Otherwise, choose the option to sync Selected playlists, artists, albums, and genres; make your selections from the lists below. When you click Apply, the iPod becomes associated with this iTunes library and your selection is synced to the iPod. Going forward, iTunes keeps the iPod in sync with this iTunes library (or selected portion of it).
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