In MWI, the future state of the universe is uniquely determined by the past state of the universe and the laws of physics. In Copenhagen, the future state of the universe isn't uniquely determined by those things, but is uniquely determined by those things plus a lot of additional bits that represent how each measurement goes. You could either call those bits part of the state of the universe (in which case Copenhagen is deterministic) or you could call them something else (in which case Copenhagen is nondeterministic), so it seems like a matter of convention. The usual convention is to call the bits something else than part of the state of the universe, making Copenhagen nondeterministic, but I don't think there's a fully principled way across theories to decide what to call part of the state of the universe.

Interesting question! My answer is basically a long warning about essentialism: this question might seem like it's stepping down from the realm of human models to the realm of actual things, to ask about the essence of those things. But I think better answers are going to come from stepping up from the realm of models to the realm of meta-models, to ask about the properties of models.

At the most basic level of description, things - the quantum fields or branes or whatever - just do what they do. They don't do it nondeterministically, but they also don't do it deterministically! Without recourse to human models, all us humans can say is that the things just do what they do - models are the things that make talk about categories possible in the first place.

Any answer of the sort "no, things can always be rendered into a deterministic form by treating 'random' results as fixed constants" or "yes, there are perfectly valid classes of models that include nondeterminism" is going to be an answer about models, within some meta-level framework. And that's fine!

This can seem unsatisfying because it goes against our essentialist instinct - that the properties in our models should reflect the real properties that things have. If water is considered wet, it's because water has the basic property or essence of wetness (so the instinct goes).

Note that this doesn't explain any of the mechanics or physics of wetness. If you could look inside someone's head as they were performing this essentialist maneuver, they would start with a model ("water is wet"), then they would notice their model ("I model water as wet"), then they would justify themselves to themselves, in a sort of reassuring pat on the back ("I model water as wet because water is really, specially wet").

I think that this line of self-reassuring reasoning is flawed, and a much better explanation of wetness would be in terms of surface tension and intermolecular forces and so on - illuminating the functional and causal story behind our model of the world, rather than believing you've explained wetness in terms of "real wetness". Also see the story about Bleggs and Rubes.

Long story short, any good explanation for why we should or shoudn't have nondeterminism in a model is either going to be about how to choose good models, or it's going to be a causal and functional story that doesn't preserve nondeterminism (or determinism) as an essence.

I think there's an interesting question about physics in whether or not (and how) we should include nondeterminism as an option in fundamental theories. But first I just wanted to warn that the question "models aside, are things really nondeterministic" is not going to have an interesting answer.

You haven't said what you mean by meaningful. It seems that we can test aspects (pun intended) of the issue... the Bell inequalities, Penroses proposed test for collapse, and so on.