I know only a little about ETCS, but I believe that the answer to your question is no. The basic reason is that even the models
of set theory themselves can differ vastly. If $M$ is a
model of ZFC, then the category $Set^M$, which is Set as
interpreted in $M$, will be a model of ETCS. But if ZFC is
consistent, then the models of set theory $M$ are diverse.
For example, some have CH and others have $\neg CH$, and
furthermore, by the incompleteness theorem, they can
satisfy different arithmetic statements. Such statements
show up in the category $Set^M$, since I believe that every
arithmetic statement (first order statement about natural
numbers) has a translation into the formal language of
ETCS. So in general they are not elementary equivalent. In
general, you can expect no nice functors between such
models, as some can have uncountable natural numbers
object, and others not, and all kinds of crazy things. In
general, it will not be possible to map the natural number
object from one to the other in any nice way.