For the US, the full name is "The United States of America". In this name "State" isn't a common noun, it has a specific legal meaning identifying one of the 13 originally independent countries that joined together to form The United States of America. I'm not sure why they were never called countries, but that's essentially what they were, and to a limited extent they still function as small(ish) countries.

This is the same as saying "The European Union" rather than "European Union".