The words mean, mesne, mien sound the same but have different meanings and spellings. Why do mean, mesne, mien sound the same even though they are completely different words?
The answer is simple: mean, mesne, mien are homophones of the English language.
To be used to convey; denote: "'The question is,' said Alice, 'whether you can make words mean so many different things'” ( Lewis Carroll).
To act as a symbol of; signify or represent: In this poem, the budding flower means youth.
To intend to convey or indicate: "No one means all he says, and yet very few say all they mean, for words are slippery and thought is viscous” ( Henry Adams).
An appearance or aspect.
Bearing or manner, especially as it reveals an inner state of mind: "He was a Vietnam veteran with a haunted mien” ( James Traub).
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.