I'm venturing an answer to this even though I don't know much about abstract mathematics in hope that it will be of some use. Please apply elevated skepticism to the below.
Scientists generally follow the lead of mathematicians on esoteric matters of mathematics, on the rare occasions where esoteric matters of mathematics concern them at all, unless it happens to be something that physicists got to first. Since ZFC is widely accepted by mathematicians, it is safe to assume that it is widely accepted by scientists (at least those few who care enough about abstract mathematics to know about it).
Science is concerned with predicting reality. This necessitates relevant mathematical axioms which define some aspect of reality mathematically, in addition to foundational mathematical axioms such as are needed by mathematics (and largely taken for granted or ignored by scientists) to define relationships between logical objects.
These more physically salient axioms may indeed be different for different predictions of different phenomena. In fact, multiple incompatible sets of axioms may underlie a single model, which still makes predictions that comport with measurements.
1: Defining axioms: An axiom is a mathematically meaningful definition of a quantity or relationship, including physical quantities or relationships. Axioms inform hypotheses, but they are not themselves hypotheses: they can be useless if hypotheses based on them predict falsely, but they don't make predictions themselves, so they can't be true or false.
2: The role of axioms in science: A set of axioms and the logical (mathematical) extrapolation therefrom constitute the mathematical formulation of a model.
3: The role of models in science: Models map past measurements to predictions of future measurements. Such a mapping between measurements, especially if a mechanism is identified, is a hypothesis. Models are discarded if their predictions of future measurements are false, kept if they are true within measurement error, and may be adjusted or retained as approximations if they are moderately close most of the time.
4: Domains, measurements, and reality: A domain is the range of all possible input measurements that a model is supposed to translate to predictions which comport with reality within a certain acceptable threshold of error. The domain may be knowable a priori or it may be discovered by measurement.
5: Axioms and domains: Axioms may define domains if the axioms themselves are based on approximations or necessarily imply a special case. For an obvious example: the axioms that underlie inviscid flow theory define a domain of inviscid flows.
6: Exceeding your domain: ...except there aren't any inviscid flows in reality. But if you apply the model to situations that are (mathematically) close to the domain, you might get useful predictions anyway. And since this is science, not mathematics, we have recourse to experiment: we can ask reality whether and how well the model works outside of its logically implied domain.
7: Effective theories: Pushing the boundary of your domain can only take you so far. Eventually your predictions and measurements will diverge. For this reason we come up with effective theories. Effective theories either leverage models known to work in other contexts, rebuilt for a new context in which the physically real mechanisms that they originally described are absent; or they mathematically mash something together out of multiple models known to work in other contexts, even if they have incompatible axiomatic foundations. In either case, unlike a theory, an effective theory doesn't purport to describe a physically real mechanism, just to be a model that successfully maps past measurements to predictions of future measurements.
8: If it works, it works: So now we're operating in a domain which is prohibited by our axioms and/or using a model based on other, contradictory axioms, which also prohibit the domain in which we're operating. But sometimes when you do this, you can reliably predict reality. Experiment, not mathematical neatness, is the ultimate arbiter of scientific truth.