If $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ are measurable spaces and there are bimeasurable injections $f \colon X \to Y$ and $g \colon Y \to X$, then $X$ and $Y$ are isomorphic. This is because the usual way to construct a bijection for the usual Cantor-Schröder-Bernstein theorem gives you a bimeasurable bijection, see this post for the construction I have in mind. The proof that this bijection is bimeasurable if $f$ and $g$ are seems to rely on the fact that $\mathcal{F}$ and $\mathcal{G}$ are $\sigma$-algebras, as opposed to more restrictive collections of sets.
Say we restrict ourselves to what you might call pseudomeasurable spaces (please tell me if there is a more common name for such a structure), where $\mathcal{F}$ and $\mathcal{G}$ are taken to be mere rings of sets (so a collection of subsets closed under binary union and set difference). Does the Cantor-Schröder-Bernstein property still hold in this case? If not, what are nice examples of spaces that mutually inject but are not isomorphic such that $\mathcal{F}$ and $\mathcal{G}$ either a: are not closed under countable unions or b: do not contain the entire set?