It seems like when I go to a cage and hit assign, only the "stray" animals show up. Is it possible to cage my pets in my zoo?
I believe you can, but you think you're only seeing the stray pets as by default it shows all of any species of animal as a single animal.
Select assign again, then hit either tab or x and it should split the list out to show every single animal, and obviously as pets are named you'll see them in the list too. Normally when assigning it just picks a random animal of the right species to assign to the cage but you want to be more specific.