Do you have a USB power adapter, like the one that comes with iPhone or other device? If so, try connecting docking cable to it, and connect the iPod there. Do you get the battery charging symbol on the iPod's screen?
Since you have both a Mac and Windows PC, was the iPod previously used with the Mac or Windows PC? In other words, do you know if the iPod is currently formatted for Mac or Windows?
Try to (intentionally) put it into Disk Mode
How to put your iPod into disk mode - Apple Support
Before connecting it, you may want to shut down (power off) the computer. Disconnect all USB devices; you can leave standard USB keyboard/mouse connected, if used. If the computer is a laptop, connect its power adapter (so it's not running on battery power). Start up computer, run iTunes, and connect iPod to a direct USB port on computer, by itself. Try more than one port, if needed.
If recognized properly, iTunes prompts you to do a Restore with a message about "recovery." Let iTunes do the Restore to erase the iPod, reinstall its software, and set it to default settings.