@jonah and I talk about this from time to time.
It’s kind of a funny thing because in frequentist statistics more often than not you see confidence intervals reported. Prediction intervals seem much less common so it’s surprising that Bayesian inference is (at least as far as I know) entirely prediction intervals.
Rob Hyndman has a nice blog post elaborating why that is (https://robjhyndman.com/hyndsight/intervals/), but the cliff’s notes are that once the parameter is modeled as a random variable, then inherent meaning of a confidence interval (i.e., 95% of all 95% confidence intervals contain the true parameter value) doesn’t make sense.
Some people (including me!) do sometimes look a frequentist properties of Bayesian intervals. That can be contentious but I view it as calibration of methods in a very practical sense.