The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and is very similar to the "multiverse".
A typical difference between the universe and multiverse views is the attitude to the continuum hypothesis.
In the universe view the continuum hypothesis is a meaningful question that is either true or false though we have not yet been able to decide which.
In the multiverse view it is meaningless to ask whether the continuum hypothesis is true or false before selecting a model of set theory.
Another difference is that the statement "For every transitive model of ZFC there is a larger model of ZFC in which it is countable" is true in some versions of the multiverse view of mathematics but is false in the universe view.