Most native speakers would recognise a distinction made by including the preposition or not...
1: She fed me leftovers
2: She fed me on leftovers
...where #1 might often simply mean she gave me some leftover food to eat on one or a few occasions - but I might also have been eating many other things (perhaps obtained from different sources).
But #2 very strongly implies leftovers was the only food I had (if it weren't for the leftovers she provided, I'd have starved to death).
Note that on is more likely than with in, say, What is your dog fed on? But it's not very idiomatic to phrase a question like that anyway (and even worse would be What is your baby fed on?).
You might ask a farmer what he feeds his chickens or pigs on/with, but the usage is a bit too "functional / commercial" for pets or babies - where you'd more likely ask something like What do you give them to eat? or What does she eat?