Here's how I see it: You're updating (approximately) over a limited space of hypotheses that might not contain the true hypothesis, and then this idea that the best model in your space can still be approximately correct is expressible both on Bayesian and on frequentiist grounds (the approximate update over models being equivalent to an approximate update over predictions when you expect the universe to be modelable, and also the best model having a good frequency of success over the long run if the real universe is drawn from a sufficiently nice distribution).

The "update" doesn't use Bayes's rule; there's no prior; there's no concept of belief. Why should we still consider it Bayesian? I mean, if you consider any learning to be an approximation of Bayesian updating, then sure, PAC-learning qualifies. But that begs the question, doesn't it?