I've not seen that terminology. I'll take your word for it, though I find it unnecessary to conflate bijection with isomorphism.
Of course we can always ask
whether <X membership_on_X> and <Y membership_on_Y> are isomorphic.
And the partial ordering of PX by the subset relation is a separate matter. Yes, of course, if X and Y are equinumerous then
<X subset_relation_on_PX> and <Y subset_relation_on_PY> are isomorphic.
But you've not shown how that serves toward proving that a countable union of countable sets is countable.
Not in ordinary set theory. X and Y are the same if and only if X and Y have the same members. X and Y may have the same cardinality, but that does not entail that X and Y are the same. Moreover that X and Y have the same cardinality does
NOT entail that <X membership_on_X> and <Y membership_on_Y> are isomorphic.
I didn't say anything about that in my previous post. But in this post I have said that yes, of course, if X and Y are equinumerous then
<X subset_relation_on_PX> and <Y subset_relation_on_PY> are isomorphic.
How that serves toward proving that a countable union of countable sets is countable is not stated by you.
Isomorphism of models (structures for a language) is one kind of isomorphism. But in set theory there is the more general notion of a isomorphism comparing (1) a set X and relations and/or operations on X with (2) a set Y and relations and/or operations on Y. If you wish to say that where there are no specified operations or relations along with said set, then X and Y are isomorphic if and only if X and Y are equinumerous, then you're free to use the terminology that way, though personally I don't find that it adds clarity to the matter.