With the word 'zijn' -- it there a way to tell whether it is "its" food or "his" food?
It's strange that there is no distinction between "The dog is eating his (a man's) meal" and "The dog is eating it's food", when both have such separate meanings.
In Dutch can you specify the 'his' in: "the animal eats his food" instead of :"The animal eats its food"?
It would need haar then.
Zijn either is used when the gender is unknown like its in English, or when it's a male.
Could it also be "the animal eats their food", as in the animal is eating the food of others?
Nope, that would be het dier eet hun eten. An alternative translation of Het dier eet zijn eten could be The animal eats his food (as in some 'him' that is not the animal).