@here_it_comes...the "star-spangled banner" rules over 50 semi-autonomous states . . . Using your logic either no anthem would serenade the American flag, or each of the 50.5 states anthems would be playable on presentation of the flag.
The US was largely homogenous up into the 1970s. National sentiments are a product of nation, not of politics. This is why
despite having a large number of states with a degree of political freedom, American national sentiment was largely uniform and embraced until only very recently. As national homogeneity has declined, the calls for changing/removing American national anthems, along with history associated with it, have risen. This is why empires do not have national anthems. Non-native nationals in an empire will only stomach being outsiders in the empire as long as they have no choice. The same is playing out in the UK now as well.
@here_it_comesAnother example is the former Soviet Union and its successor, the Russian Federation, the latter of which is 21 or 22 states depending on your stance on Crimea. All those states, to the best of my knowledge, unite under the "Gosudarstvennyy gimn Rossiyskoy Federatsii" or "State Anthem of the Russian Federation".
The USSR was an empire and it's original anthem, the Internationale, is not a national anthem in the slightest, it is communist political anthem. The anthem that replaced that was meant to bolster the fighting spirit among Russians during WW2, as national sentiments touch the heart more than a political anthem. And even then reading the lyrics, it is overtly an imperial anthem and tosses the nationalist sentiment a bone for effect. Modern Russia is in a strange state of returning to a national spirit, while still having to maintain the spirit of citizens who lived in the old empire and their anthem reflects it, enshrining the state in the first line. Their current anthem is national in word choices, but not much more. It's poor popularity proves this.
Similarly, and a little closer to home, and also closer to where we are in game with the three superpowers, the European Union has adopted "Ode to Joy" from the final movement of (Ludwig van) Beethoven's 9th Symphony as the overarching anthem of the union, yet all the memberstates retain their individual anthems such as France's "La Marseillaise" by Claude Joseph Rouget de Lisle or Germany's "Deutschlandlied" their "Song of Germany".
This is precisely how the superpowers would pick their anthem. As a matter of formality, calculated towards desired political aesthetics and notions. No different than how songs are picked for political rallies. The European nations maintaining their anthem despite the EU has no relevance to ED superpowers, as the EU is not a formal empire ruling over the European nations. Should that come to pass, those national anthems will be expunged yesterday.
So I see no reason why the individual planets / colonies / even settlements couldn't have their own identity, but unite under their corellating superpowers anthem?
A people that form a nation will necessarily have a sense of identity strong enough that they will want to self govern. This is where movements to self-determination stem from. Nations don't want rule by foreigners, which is precisely what an empire is, and rulers of foreign subjects don't want their subjects trying to break away. So, you may see no problem in some backwater planet having a strong national identity, but the empire that rules over them certainly would, guaranteed. Their national anthem may sound like interesting culture to you, but to some Federation Governor, it means revolution if isn't dealt with.
It's not hair splitting. The differences between an empire and a nation, and the products of either, are so far apart they are practically opposites. Empires will never have national anthems, not for long at least.