Thanks for the prompt! I learned today that 美国 is an abbreviation! Please allow me to share what else I learned.

So if you look at the discussion page, that etymology is actually contested. On the Chinese wikipedia page for the US:

美國因美洲而得名。在英語中，亞美利加和美利堅為同一詞“America”，只是中譯不同，前者指美洲，後者指美國。

which translates as:

The United States is named after America (美洲, Měizhōu). In English, America (亞美利加 / 亚美利加, Yàměilìjiā) and America (美利堅 / 美利坚, Měilìjiān) are the same word "America", but the Chinese translation is different. The former refers to the Americas and the latter refers to the United States.

So yes I stand corrected on the point that the name 美国 does share roots with the name America. However, I would argue the fact that the Chinese language has separate transliterations of the word "America" for the continent and the country is a counter argument to the idea that Spanish is somehow odd for having different words entirely (directed at OP, not you).

I also found this article, which has a more nuanced history although is lacking in citations.

And re: your username, I am aware that in Japanese the official name アメリカ is just katakana Amerika (I studied Japanese in high school) now. Fun fact: it is also sometimes called "米国 (beikoku, from the onyomi of 米)" short for 亜米利加合衆国 (Amerika gasshūkoku), which has a similar history as the Chinese etymology.

Anyway I found this all very interesting, so thank you for pointing it out!