Hey tcwjohnson !
Yes, you should be able to program the nRF9160 Feather using the nRF9160-DK. (It’s the recommended way to do so)
Can you show how you have it connected? Is the nRF9160 Feather powered via USB when you’re programming? (that’s required here) I personally short SB24 which forces the nRF9160DK to use the external programming port. If SB24 is shorted, you don’t necessarily have to power the nRF9160 Feather externally.
Otherwise you may find yourself accidentally programming the onboard nRF9160 if the connection is not so great (or if you forget to connect the cable, etc, etc.)
Hope that helps!