QFT and most of the interpretation issues in QM are only marginally relevant to answer your question.
The key point is the role of the wavefunction in the theory and how the theoretical descriptions are connected to the real world.
The second point is quite simple but not trivial. Theoretical concepts are how we describe and understand the world, and there is no "deeper" alternative. We can never say what a phenomenon is. We can just build concepts to model as faithfully as possible what happens around us. To be precise and quantitative math plays an important role, but it is an auxiliary role. The bare fact that we have a mathematical description of some phenomenon does not imply that such a description is totally disjoint from what can be measured. Instead, using mathematics is the way to allow quantitative measurements.
When we have a mathematical model which uniquely maps measurable phenomena into mathematical objects, we can use the latter as proxies of the former.
Things get different if there is not a unique mapping. In such cases, we say that a mathematical quantity of a physical theory is not measurable. And consequently, we assign to it a reduced physical meaning.
The simplest example I can imagine is the case of energy. Our theories always introduce energy in the form of differences in energy. Therefore, variations of energy are measurable while absolute energies are not.
Slightly more complicated, but closer to wavefunctions, the case of the electromagnetic (em) potentials in classical electrodynamics: em fields are directly measurable, while there is a one-to-many relation with the em potentials (gauge invariance of the theory). Once again, the missing one-to-one correspondence between phenomena and concepts of the theory makes the classical scalar and vector potential non-measurable.
The case of the wavefunction in QM is similar. It is a concept required to describe what we measure, but there is no unique correspondence with the physical situation.
Within the wave formulation of QM, the non-uniqueness has two origins. On the one hand, all the observable consequences of the theory do not use the wavefunction directly but only quantities defined within an arbitrary phase-factor, analogously to the previously cited examples from classical physics. On the other hand, the time dependence of the observables may be entirely assigned to the wavefunction or the operators or even shared between operators and wavefunctions (the cases of the Schrödinger, Heisenberg, and Interaction pictures). The possibility of such choices further increases the non-uniqueness of the wavefunction.
In conclusion, the question about the physical meaning of formalism is always a matter of establishing a unique correspondence between concepts and measurements. However, when a theoretical concept fails to have a unique link with measurements, we consider it not directly physical but an auxiliary concept.