The lack of a universally agreed upon definition does imply the lack of an objective definition. And it also does not imply there isn't a general definition that most people agree on that works in most cases.

Almost all words have different meanings when you get into a specific niche. "Elastic" in casual contexts means stretchy, but in physics contexts, specifically describes a materials ability to return to it's prior state after being stretched. But if I said "Elastic" lacks a definition in physics entirely, we wouldn't run around saying "nothing is elastic".

Yet for some reason people do that exact thing with Vegetables/fish.

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.