Deanna consistently referred to herself as a Betazed, even with her mixed Betazed/Human ancestry, I can't remember Deanna ever personally calling herself a "Federatista" or some such.
I'm pretty sure when Deanna refers to herself as Betazed, she's referring to her species/race/heritage, not her government. No one ever calls himself an Earthling or Terran (except in the MU), it's always human.
When referring to nationality, it's Federation.
The nature of the UFP is never fully stated onscreen, and we've only seen snippets of the Federation Constitution and Charter, but if I get the feeling that's it's a level of government somewhere between what the EU is and what the US is, at least by the 24th century.
The biggest evidence of that is Starfleet itself, which is a unified interplanetary defense/exploration force. It's not just a coalition with troops from different planetary militaries.
And when Changelings infiltrated Earth, was the Earth government ever even mentioned? No, Sisko goes to the Federation president with what he's found, and Starfleet ups security on Earth - not just in Paris and San Francisco, but in New Orleans.
The last time we hear anything about an Earth government is mentions of UESPA early in TOS, before GR even conceived of the UFP.