Not sure whether you have sorted this now, but i had the same problem (hence reading this thread) and spoke to Apple Support on the phone about my ancient iPod Touch. After trying mulptiple times to sign in to my iCloud account, the Touch was requesting a Recovery Key, which like you, i could not get, or have the option to get.
In the end, it was a case of deleting the iCloud account i was trying to sign in with (hence needing the recovery key) then re signing in. At this point, when signing in again my phone/ipad was triggered to give the two factor 6 digit numerical code. What i didn't know and the helpful chap on the phone told me, is that eventhough the older devices do not get the 'pop-up' box for said six digits, where the password is required for the apple id, you then add the six numbers straight after your password with no spaces.
Worked for me and no longer asks for a Recovery Key.