What do you mean by objective? If i say that vegetables are things contained in this list: {carrot, potato} and nothing else, wouldn't this be an objective definition of my own making?

If i say that fish are all members of the monophyletic group of vertebrates, including hagfish, reptiles, birds, and humans, but not including tunicates, that is also an objective definition.

I think you mean that these things don't have a universal generally-accepted definitions, as reasonable people might use "fish" and "vegetable" to mean very different things in different contexts.

They can still be valid and useful definitions, it is still useful to talk about eating a healthy diet of fruits and vegetables, or seeing fish at the aquarium, but the definitions are fuzzy until you specify them for a specific context.