On your edit, it's probably your wording of the issue, it's somewhat muddied what you try to say. It's a controversial issue and people get picky, so if your phrasing is a little off you're easily interpreted as being wrong. But I'm thinking one way you could think of the constituent countries as sovereign nations is this:

Once, there were several countries who decided (circumstances, ignoring those) to work together. They formed a joint council that they would all contribute to, and all obey. A government of governments, of sorts. A country of countries. Neither nation gave up their identity or their sovereignty, but over time it became convenient to sort of ignore it in many matters (like passports). They're sovereign not because they act sovereign, but because they never really stopped. It's hard to find a point in time where you can say "2 months ago these were separate nations, but today they are one". They just kind of float in a gray zone.