To follow up: How recent of an invention are national anthems? Seeing as standardized national flags are more recent than a casual observer would believe, does the same rule apply to anthems?
In short, most anthems conform to the Western pattern, but a select few don't for deep-seeded ideological and cultural reasons. Because of this, how a nation's national anthem sounds actually tells you a lot about the nation's history. Anthems were a Western idea originally, the first adopted in the 1500s in the Netherlands. Spain and the United Kingdom, among a handful of other countries, followed suit in the coming centuries. However, national anthems proliferated during the age of nationalism, and the majority of European anthems were composed and instituted in the 1800s. South American nations adopted anthems at around the same time - unsurprisingly, the former colonies adopted musical styles very similar to those in Europe. This pattern would be repeated in Africa in the 1960s for similar reasons related to colonial influence.
However, the proliferation of Western musical styles in Asia is a fascinating topic, and there was a wide variety of anthems adopted originally. Anthems were an imported idea from the West, so most Asian countries did not adopt them until the late 1800s or early 1900s. They were composed by native composers for the most part, under the direction of the government. Therefore, how they sounded depended largely on the philosophy of the government. This is in stark contrast to European countries, where anthems were often composed long before being adopted by the governments, and reflected a common European musical score.
These government-commissioned anthems in Asia are some of the most unique anthems produced, and incorporated local melodies to emphasize cultural traditions. Some examples:
Mecidiye Marsi - Anthem of the Ottoman Empire
Salam e Shah - Anthem of Persia
Cup of Solid Gold - Anthem of the Qing Dynasty
The above three anthems, produced by countries facing a clash between Western and traditional ideas, embody the attitudes of these countries towards Westernization. The Ottoman and Persian anthems incorporate Western instruments and were meant to be fully convertible to Western orchestral score, but still incorporated local musical styles. This reflected the ambivalent attitudes of their governments towards Western ideas - recognizing the superiority of Western technology and some Western traditions, but still attempting to cling to their own culture and history.
The Chinese anthem, in contrast, incorporates almost no Western instruments, and reflects the Qing's adamant insistence on a "Chinese essence" in all things, and on the basic superiority of their country. In stark contrast, the present Japanese anthem, written during the Westernizing reforms in the 1800s, conforms almost perfectly to Western models, and reflects that nation's wholehearted embrace of Westernization.
Come the age of Communism, a new tradition of socialist national anthems proliferated - alternatingly called "socialist music", "mass songs", and "revolutionary music" by historians. Like the "ethnic" national anthems of the Ottoman, Persian, and Chinese empires, revolutionary music was state-sponsored and directed. It was meant to reflect the socialist character of the countries involved, and rejected "bourgeois" Western melodies. Some examples:
National Anthem of the Soviet Union
It should be noted that not every Socialist country chose to compose a new national anthem - China and the Soviet Union did primarily because their governments wanted to emphasize Socialism as one of the deepest parts of their national character at the time of adoption. With the exception of the DDR, the Soviet Union's client states in Eastern Europe generally decided to adopt older songs, or to compose new songs based on older styles. It should also be noted that the Soviet Union's first anthem - the Internationale, composed by a Frenchman in 1871 - fit most of the musical patterns of other Western anthems.
TLDR: Most anthems do fit the Western model because of Western influences, anthems being a Western European invention to begin with, and the pre-eminence of Western instruments. Some do not - these exceptions were usually chosen by governments to convey ethnic or ideological differences with the Western world.
Sources:
Krebs, Stanley D. Soviet Composers and the Development of Soviet music.
Khaleqi, Ruhollah. The story of Iranian music.
Eccles, Lance. Anthem for a Dying Dynasty: The Qing National Anthem through the Eyes of a Court Musician.
Howard, Joshua. Music for a National Defense: Making Martial Music during the Anti-Japanese War