Exchangeability is often interpreted as the Bayesian version of i.i.d assumption . In i.i.d assumption we assume that the parameter we are interested in exists in some meaningful sense and the data at hand (1 or $n$ observations does not matter ) were generated as independent trials conditional on the parameter .
Exchangeability is the condition that the joint density of the data remains the same under re-ordering or re-labeling of the indices of the data , so if we assume for example a binomial model then under exchangeability two sequence of random variables , each with the same length $n$ and the same proportion of ones would be assigned the same probability only the number of ones matters not the location of the ones .
So that the assumption of exchangeability in Bayesian inference means we believe that the data are exchangeable then it is as if there is a parameter (say $\theta$) that derives a stochastic model generating the data and a density over $\theta$ that does not depend on the data , this density interpretable as a prior density thus the existence of a prior density over a parameter is a result of the exchangeability rather than an assumption
Edit based on Xi'an's comment :
Some models allow for exchangeability and others do not .In the hierarchical models also we can not consider the entire sequence of data as being exchangeable instead the data within any given group might be considered exchangeable that means we condition on the group and that is so called conditional exchangeability