The supported device for iOS 13 as follows:
iPhone 11 Pro
iPhone 11 Pro Max
iPhone XS Max
iPhone 8 Plus
iPhone 7 Plus
iPhone 6s Plus
iPod touch (7th generation)**
But checking the fine print with ** for the iPod touch states that it ...
iPod Touch 7th generation is supported by iOS 13. Sometimes software update is slow to catch because of caching or similar behavior. Try restarting it and go to
Settings -> General -> Software Update
on the device, it should work.
In the picture you can see a restore image for an unreleased version being available for ipod touch 7th Gen