There are some special cases in which a frequentist 95% confidence interval and a Bayesian 95% credible interval based on some sort of default prior are numerically the same, but that doesn't happen in general.
Statisticians would hardly have been vigorously debating the issue for two centuries if it didn't really matter.
>>>> Let’s now suppose that we’ve done a Bayesian analysis. We’ve specified a prior distribution for the parameter, based on prior evidence, our subjective beliefs about the value of the parameter, or perhaps we used a default ‘non-informative’ prior built into our software package.
At first blush the difference is that the Bayesian is using more information. Now don't get me wrong, if Bayes theorem and its progeny give us useful tools for incorporating that information in our analyses, so much the better.
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.