Why are most countries nations, even when they have multiple ethnic groups with national aspirations ? For e.g. Russia.
I believe as Pierre Elliot Trudeau did, the nation must not be the basis of the state
No, not all countries are nation states. Most (possibly all) large countries have started out as multinational states.
India is the best current example of a highly multinational state, recognizing and governing itself as such, with two language families and hundreds of distinct ethnicities. While Hindi is the most prominent, it remains a simple majority rather than an assimilation policy.
Some multinational states have managed to merge or convert multiple ethnicities into a single majority national identity. China, originally a diverse empire, has over centuries culturally assimilated most of its ethnicities into the Han group.
Bosnia and Herzegovina is perhaps the most clearly multinational state in Europe, reflected in its name. The United Kingdom consists of England, Scotland, Wales and Northern Ireland, each with their own language, but downplays the distinction sometimes. Switzerland has a canton-based identity.
Spain is an example from the other end of the spectrum, a de-facto multinational state that refuses to recognize itself as such, and it had internal conflicts over its lack of regional identity policies.
The Soviet Union was a conglomeration of multiple multinational states. The population was deliberately mixed up as a matter of internationalist ideology, so even originally monoethnic provinces were diversified. After it broke up, some states stayed multinational, others tried converting into nation-states, creating a series of conflicts.
The US is highly diverse and started out as multinational. It has assimilated its various European settlers, English, German and Spanish, into a new identity as Americans, though that hasn't always worked out for other cultures.
Its status as multinational or nation-state is disputed, but that's an academic distinction, as the US abstracts ethnic differences to an unusual extent. Rather than internationalism, the US unity has more of a transnationalist foundation. For practical reasons, its governance system is typical of multinational states.
For the most part, the distinction between a nation-state and a multinational state is made predominantly on whether the acting government recognizes any identity rights (language, autonomy, representation) for minority nations and ethnicities, rather than their proportion and the extent of distinction between them.
This may be lost in translation, but the Russian word nationality ("национальность") is best translated into English as 'ethnicity.' So the proper way to call Russia (at least the Russian Federation) would be a "multi-ethnic state" rather than a "multi-national state." Although the USSR could have been called a "multi-national multi-ethnic state" because each "republic" of the USSR had partial sovereignty (for example, it had control over the republic's language).
In the US, citizenship and nationality are not just synonymous, but also legally indistinguishable concepts. For example, US passports list nationality of all citizens as "United States of America."
So "nation state" is not meant to indicate that the state is ethno-centric, but rather to indicate the full sovereignty of the entity. This is sometimes necessary in order to distinguish a fully-sovereign entity from a partially-sovereign "state." All "states" of the United States have fairly strong partial sovereignty because of federalism. For example, no state governor can be hired/fired/appointed by any official of the Federal government (not even the President). In fact, the President can't even fire any state's police officer because police are employed by states (but there are also Federal law enforcement agencies). There are many other examples.
Having said that, the United States, as a whole, is a nation state. The defense of the entire entity of the United States, for example, is referred to as "national defense" in the US constitution.
In other words, using both words "nation state" is just a disambiguation in English.
You won't get much push back, because you'll be using words which have meaning in the context of discussing states. But if you tell someone that a country is a "multi-national state," they will just assume that you mean it has a federalist governing system. They may not know what the word "federalism" means, but they will assume (from their daily experience) that this is what you mean if you use the term "multi-national state." They will still think that you are trying to push some agenda though. The only word people would expect to hear after hearing "multi-national" is the word "corporation."