I am definitely not the person to go to for this question, but my understanding of a nation state is one where everyone shares a common cultural identity, which doesn’t seem to be the case in Westeros?
Westerosi define themselves more regionally, by kingdom, like westermen and northmen and rivermen. The Neck also represents both a geopolitical and cultural divide between North and South, with northmen sometimes feeling disdainful of southron culture and vice versa. The people are also ethnically divided, with the primary ethnicities being Andals, First Men, Rhoynar, and (a relatively small number of) Valyrians. These cultural divides represent vast differences in religion, customs, regional accents, etc. and it’s one of the reasons why I think the Seven Kingdoms will break apart in #endgame.
As you suggest, I believe Westeros might technically be called an empire, but I think Aegon the Conqueror eschewed the title “emperor” because that title wasn’t working out too well for other dragonlords after the Doom.
You might want to check with someone like @racefortheironthrone or @warsofasoiaf though, because they would be able to give you a much better answer, and I’m not sure if I’m correct.
@joannalannister raises an excellent point, with regards to Westeros’ lack of a common national identity.
But to chime in on the state part, Westeros is largely a weak feudal monarchy and not a state that can honestly claim to exercise unitary, legitimate and unchallenged authority over its territory: the regional nobility of Westeros just pulled down one dynasty and are bid fair to do it to a second.
The state doesn’t really have a centralized bureaucracy that can raise the necessary revenues for a standing army or keep adequate records on the population, it doesn’t have a centralized judicial system, it’s rather underdeveloped when it comes to national infrastructure, and so on and so forth.