The iPod 5th gen is 2012 technnology and has similar hardware architectures to the iPad 2 and 3, which has also been discontinued.
The iPad 2, 3, 1st generation iPad Mini and the 5th Gen iPod Touch are all ineligible and excluded from upgrading to iOS 10.
They all share similar hardware architectures.
A less powerful 1.0 Ghz CPU that Apple has deemed insufficiently powerful enough to even run the basic, barebones features of iOS 10
The 5th gen iPod Touch CPU is a clocked down 1.0 GHz to 800 Mhz and, like the iPad 2, only has 512 MBs of RAM.
Why would you want to upgrade to iOS 10, anyways?
Since iOS 8, older iDevices have only been getting the most basic of iOS features.
Older iDevice users get none of the newest and coolest cutting edge features that newer model iDevices, with better hardware specs, can handle with more aplomb.
The same older iDevice. users who whined and moaned and groaned that both iOS 8 and 9 slowed down the performance of their iDevices would come back, once again, in angry droves, with even more complaints of slowdowns under iOS 10.
Plus, the initial release of iOS 10, at the moment, seems riddled with bugs and undocumented feature omissions that current iOS upgraders are, currently, struggling with.
FYI,
Just because your iPod Touch 5th gen is no longer able to upgrade to the latest iOS doesn't mean youri iPod Touch is suddenly useless.
Your iPod Touch 5th gen will work as it always has and the apps you have installed on it will continue to update and receive some level of app updates relevent to your current iOS.
Most apps have a minimum iOS version they run under, anyways.
You have had four years of iOS upgrades and updates.
Your iPod Touch 5 will still do everything it does now and you will, get, at least, another full year, perhaps two, if you decide to you want to wait longer to purchase a new iPod Touch model.
I just needed to put that out there for you.
Your 5th Gen iPod Touch is NOT suddenly becoming obsolete.
Good Luck to you!