Let $A,B$ be sets and let $f: A \to B$. Then we say that $A,B$ are isomorphic under $f$ if $f$ is a linear function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are homeomorphic under $f$ if $f$ is a continuous function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are ($C^{r}$)-diffeomorphic under $f$ if $f$ is a $C^{r}$ function with $C^{r}$ inverse that maps $A$ onto $B$ in a one-to-one manner; and so on.
I think it may be convenient to have a name for a pair of sets between which there is a "simple", need-not-have-additional-property bijection, for if not then one may have to use many words to articulate.
I am not aware if there is already one such name, so I would like to know it if any exists and, if no such name exists, I would like to solicit some ideas for coining a new terminology, say simply calling such a pair of sets morphic under the bijection.
Probably the most standard term of the sort you're looking for is "equipotent", though it isn't used particularly often (more often, people will just say two sets "have the same cardinality" or "are in bijection"). People who think about categories a lot sometimes say "isomorphic as sets".