Most students know that kanji names exist for Japan, China, and Korea. More countries have kanji names though, most of which are purely historical but are sometimes used as shorthand for the country. More advanced speakers will recognize 仏蘭西 (フランス) for France (shorthand 仏), 英吉利 (イギリス) for England (shorthand 英), and 亜米利加 (アメリカ) for “America” (US) (shorthand 米).
From these shorthand kanji, we can create new words using the on’yomi, some of which many learners already know: 英語 (えいご), 仏語 (ふつご), 米国 (べいこく).
One big usage for place name shorthand is what I would call relational words, which we would formulate as “A–B” in English. For example, Osaka (大阪)–Kobe (神戸) becomes 阪神 (はんしん) in Japanese (cf. Hanshin Line).
Note how each name lends one kanji, and the whole word becomes read using on’yomi.
My favorite, as someone who’s from Hawaiʻi, is the [very, very obscure] word for Japan–Hawaiʻi (布哇), 日布 (にっぷ). One of the largest Japanese-language newspapers in Hawaiʻi used to be called the “Nippu Jiji” (日布時事).
Although many consider full kanji names (especially of western countries) to be pedantic information, note that encyclopedias like Wikipedia do use shorthand kanji to refer to languages (see: https://ja.wikipedia.org/wiki/%E3%83%91%E3%83%AA?wprov=sfti1, which uses 仏 as short hand for フランス語), and Japanese geographical and diplomatic terms like 日米, 日英, 日仏 are still used, especially in newspapers and formal documents.
For having made it this far, enjoy this video of calligrapher Takumi writing the names of a bunch of countries in kanji: https://youtu.be/0me17UV8_7I