What is the function of це in this sentence? Would "Кінь - тварина" also mean "A horse is an animal"?
Yup, «Кінь — тварина» is the same.
«Це» in Ukranian ( «это» in Russian) is typical for definition-like sentences of A is a B structure, whether objective ("A field is a quantity that has a value defined in each point in space and time ") or metahporical ("Hope is life").
The rule is not very obvious. If the word "це" is missing than the dash is required. But if predicate is nominative noun than dash is also required. But this rule is complicated and often is broken
Actually, the rule is quite clear: the dash replaces the auxiliary verb 'is'.