The Mandarin dialect of the Chinese language has a set of phonetic characters, called Zhu Yin (注音) that can sound out every possible character. So dictionaries pre-CCP are organized using the phonetic set. The idea is probably that if you can speak the language and know the phonetic set, you can find the character. Also Mandarin was used because it was the most commonly and widely spoken / understood dialect.

The post CCP dictionaries that I used from China when I was a kid were organized according to Pin Yin (拼音). The CCP mapped the phonetic characters to the English alphabet, and the characters are organized according to alphabetical order based on each word's phonetic representation. The idea is the same, if you can sound out the character, you can find the character.

There is often also a prefix in the dictionary that has all the characters organized according to the "stem" of the character, we call 部首, then the remaining strokes to complete the word. Each character's page is listed in the dictionary. That way, if you run into a word you don't know how to say and don't know its meaning, you can look it up by "stem" plus stroke. See pic below.