There is an argument in English usage that one should hyphenate so-called compound adjectives as in Mexican-American War to indicate that both adjectives are of equal import. This is less common nowadays because one big usage, hyphenated identities, began to be seen as pejorative. Calling someone an Irish-American politician could specifically imply they were less than wholly American, whereas an Irish American politician could be more easily read as an American politician of Irish extraction. You can see the sentiment surrounding so-called hyphenated Americans famously in this 1915 speech by Theodore Roosevelt, though my OED traces "hyphenated" in this context to 1893.

To return to your question, though, there were too many belligerents to reasonably call the Korean War simply Korean-American War, and without a second adjective, where would the hyphen even go? One could argue the lack of hyphen shows the increasingly complex and global nature of such conflicts.