An SSD in the 1.8-inch hard drive form are typically designed (and tested) to be used (as replacement) in small laptop computers that used the same type of mini hard drive as 5th gen and later "classic" iPods. They may not work reliably in iPods. For example, the iPod may not supply enough power.
If you can get the iPod to appear in iTunes, on the iPod's Summary settings screen in iTunes, set it to Enable disk use (if it's not already), and click Apply. Does having disk use enabled make a difference in being able to sync songs?
Another thing to try is to use Disk Utility to Erase (reformat) the drive once, before doing a Restore in iTunes. Quit iTunes, if running. If the iPod does not appear in Disk Utility when connected, use the Disk Mode procedure first
Putting iPod into Disk Mode - Apple Support
and then connect iPod. In the Disk Utility sidebar, select the iPod device, not the volume indented below the device. If you are using OS X El Capitan, with its "new look" Disk Utility, click Erase. In the Erase settings, for Scheme, select Apple Partition Map (not the default GUID Partition Map). For Format, select OS X Extended (Journaled). For Name, it does not matter. Click Erase. (If your OS X is older, please post back.)
If Erase completes successfully, quit Disk Utility and run iTunes. In iTunes, do a Restore on the iPod to install the iPod's onboard software. Then, try syncing again. If the Erase errors out, or the the problem is not resolved after doing the Restore, you may want to consider an alternative that uses commonly available (and now quite affordable) flash-based storage in the form of compact flash card or SD card, on an adapter. This web site has relevant and interesting information
https://www.iflash.xyz/
These adapters are designed specifically for use with iPods (5th gen and later). I have not used these particular adapters with 5th gen and later "classic" iPods, but I have done similar mods on older 3rd gen iPod, 4th gen iPod, and iPod mini (using different adapters). They work well with flash-based storage instead of mini hard drive; they actually work better now, with more storage capacity.