[Gondolin]

Only speaking about arithmetic rather than full set theory: in (full) second order arithmetic theory all models of the natural number N are isomorphic (in a unique way). The big drawback is that this theory is completely undecidable, so we usually interpret it in first order logic set theory where it becomes incomplete. By Godel, you cannot have an arithmetic theory with decidable proofs and completeness.

In set theory one has to be careful that you can have models which are not transitive models. So Löwenheim-Skolem implies that if you have a model, you have a countable model. But you may not have a transitive countable model. However you do have a transitive version of Löwenheim-Skolem, by Mostowski collapsing lemma: "there exists a transitive model" <=> there exists a countable transitive model".