An surprised view:
Uncertainty quantification does not strictly require a hidden probabilistic view, but probability is the most common language for it.
A cleaner way to say it:
UQ needs a way to represent limited knowledge, variability, or error. That representation may be probabilistic, but it does not have to assume some hidden random truth behind everything.
Several views:
This is the standard approach. You assign probability distributions to uncertain quantities:
[
$\theta \sim p(\theta), \quad y \mid x,\theta \sim p(y\mid x,\theta)$
]
Then uncertainty is expressed through posterior distributions, predictive intervals, credible intervals, confidence intervals, etc.
This is natural when uncertainty is treated as randomness or incomplete knowledge. Bayesian UQ especially uses probability to represent epistemic uncertainty: uncertainty due to lack of knowledge.
But this does not always mean there is a literal “hidden dice roll” in reality. Probability can be interpreted as:
[
$\text{degree of belief}$
]
rather than objective randomness.