I have not tested this with many animals, only rimdogs. It is pretty consistent with rimdogs though. If you have some dogs with no zone restrictions, and a stockpile filled with potatoes, the dogs will starve to death.
Perhaps it is intended that dogs dont eat potatoes, but the wiki says otherwise (http://rimworldwiki.com/wiki/Omnivore)
More likely it is a special case of bug 2091 (https://ludeon.com/mantis/print_bug_page.php?bug_id=2091)
I am playing with the SK modpack and have not bothered to test it in vanilla. Please forgive me if the mods turn out to be the cause
Animal diet has been reworked in A13.