Yes, countries in the Sinosphere have historically used Chinese characters to write their languages. That's why Korean "yaksok" and Japanese "yakusoku" sound so similar. Both words are written with the same Chinese characters, "約束". The characters were borrowed from Chinese, but each language adapted them to its own pronunciation system.

For example, "library" is pronounced "tu-shu-guan" in Chinese, "do-seo-gwan" in Korean, and "to-sho-kan" in Japanese. All three can be written with the same characters, "圖書館". In modern Korea, though, people use Hangul, so very few Koreans actually know how to write "library" in Chinese characters. In Japan, Chinese characters are still heavily used, but for difficult ones, they often write kana alongside them as a reading aid.

It's very much like how Latin "universitas" became "university" in English, "universidad" in Spanish, and "università" in Italian.

There's a significant amount of Japanese loanwords in modern Korean due to Japanese annexation(1910-1945/1965), as well as in modern Chinese to much lesser extent.

These aren't an indication of a shared vocabulary or ancestry, just loanwords for concepts that were novel and scientific by victorian standards.

yes, and you will find a lot of borrowed words from chinese (and sometimes japanese) sound similar (more or less) in both languages

a big one: hanja (kr) kanji (jp) both are 漢字