In short: They both rely on a "background model" that you are implicitly working in. The "background model" and the model in discussion then disagree on which sets are countable.
That is e.g. you fix some model (perhaps a model that satisfies ZFC). Then you work within that model to create a submodel .You can then describe the submodel using properties of the outer model. In fact it turns out that you can have submodels of ZFC that are proper sets in the outer model, i.e. a single set can contain an entire subuniverse of sets that themselves satisfy the entirety of the ZFC axioms.
Using the outer model you can then talk about global properties of the submodel.
In the case of the Löwenheim–Skolem theorem it turns out that you can have a submodel satisfying the axioms of ZFC that can have arbitrary infinite cardinality in the outer model. In particular you can have a submodel of ZFC that has only a countably infinite number of sets as measured by the outer model. And in fact every element of that set can also be a set of countably infinite size according to the outer model.
According to the submodel there is no notion of the cardinality of itself, since ZFC does not have a set of all sets. Likewise the submodel "thinks" many sets within it are countably infinite. This is how the submodel is able to still satisfy ZFC.
However, the outer model disagrees with the submodel and instead thinks that the submodel is "impoverished." The submodel is missing the functions that it needs to "realize" that bijections exist between certain sets. These functions exist in the outer model.
That is e.g. you fix some model (perhaps a model that satisfies ZFC). Then you work within that model to create a submodel .You can then describe the submodel using properties of the outer model. In fact it turns out that you can have submodels of ZFC that are proper sets in the outer model, i.e. a single set can contain an entire subuniverse of sets that themselves satisfy the entirety of the ZFC axioms.
Using the outer model you can then talk about global properties of the submodel.
In the case of the Löwenheim–Skolem theorem it turns out that you can have a submodel satisfying the axioms of ZFC that can have arbitrary infinite cardinality in the outer model. In particular you can have a submodel of ZFC that has only a countably infinite number of sets as measured by the outer model. And in fact every element of that set can also be a set of countably infinite size according to the outer model.
According to the submodel there is no notion of the cardinality of itself, since ZFC does not have a set of all sets. Likewise the submodel "thinks" many sets within it are countably infinite. This is how the submodel is able to still satisfy ZFC.
However, the outer model disagrees with the submodel and instead thinks that the submodel is "impoverished." The submodel is missing the functions that it needs to "realize" that bijections exist between certain sets. These functions exist in the outer model.