The result of Bayesian inference is a whole posterior distribution, not just a confidence interval. Any attempt to produce a frequentist version of a posterior distribution is either going to end up just being Bayesian inference in disguise, or be inconsistent.

But even if we focus just on confidence intervals and credible intervals, there needn't be the equivalence you state. A comment elsewhere here discusses a ridiculous confidence interval that is either the whole parameter space or the empty set. That's never going to be what a Bayesian credible interval gives you.