In short, a state is about turf, and a nation is a people, and you need them both to look similar on a map to make a nation-state.
This is mostly a language confusion for non-native English speakers. Nation, country, state, a people, nationality, ethnicity, citizenship etc. are used in confusing ways for speakers of other languages.
For many, "nation state" just means an independent state (roughly speaking, a UN member, note also that the UN is called United Nations), because just saying "state" could mean a subdivision, such as a US state. And "country" can be confused with the subdivision of the UK (they call, e.g. Scotland a "country").
In more precise contexts of political history, "nation state" mostly refers to modern (post-World War I) countries that more or less correspond to a people speaking the same language and having the same ethnic identity. It delineates nation states from the previously more common multi-ethnic empires and kingdoms, such as Austria-Hungary or the Holy Roman Empire etc.
Similarly, in English, nationality is often an exact synonym for citizenship, while speakers of other languages expect it to mean ethnicity, e.g. an ethnic Hungarian in Romania with Romanian citizenship would be considered a "Romanian national" in English-language news. This often makes people confused/angry. Also, in some contexts in English, "ethnicity" is more like a euphemism for something like "race", but not quite (e.g. in the US "Latino" is considered an "ethnicity" but not a race). In that sense "Hungarian" would not count as an "ethnicity" at all, but still phrases like "ethnic Slovak" refer to a minority group in a different country than Slovakia. But also "ethnic" can also just mean with "exotic foreign origin", e.g. "ethnic food" or "an ethnic woman" (this was really weird when I first read it). But I digress.
I've never heard Belgium as a stand-in-for-Brussels-as-a-stand-in for EU.
> ... because just saying "state" could mean a subdivision, such as a US state ...
I would very much like to record phone calls made by me.
When the company on the other end denies what we agreed a recording would be useful.
Note that this doesn't mean that a state with multiple ethnicities/languages can't be a nation state. Indians, for example, generally have a clear national identity, despite being citizens of a huge federal republic with dozens if not hundreds of languages spoken, some of which don't even share a common language family. So, India is a nation state, unlike Belgium.
However Walloon people definitely feel Belgian and have a Belgian pride.
Also, i don't know if a nation state is defined by having a national identity?