The useage of dB in the case of probability is exactly as the 1 Bel -> 10x meaning, but that is the technical meaning, without further context of the unit in the logarithm. There is no need to conceptualise this further as involving power or amplitude (which does not apply in probability).

I think that the unit having been popularised in the telecommunications industry just meant that every other instance of a log_10 ratio in physics lead to a realisation that it was a Bel. For Bayesian odds, this was probably because even the development of Bayesian probability was largely advanced by physicists (E.T. Jaynes being a famous example), who also were trained, and often worked, in signal processing of some kind or another. But I doubt they would have thought about this "power-ratios-only" adherence that is more the conception of telecommunications engineers, as opposed to physicists.