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.