The words me, mi sound the same but have different meanings and spellings. Why do me, mi sound the same even though they are completely different words?
The answer is simple: me, mi are homophones of the English language.
Used as the direct object of a verb: He assisted me.
Used as the indirect object of a verb: They offered me a ride.
Used as the object of a preposition: This letter is addressed to me.
Informal Used as a predicate nominative: It's me. See Usage Notes at be, but, I1.
Music The third tone of the diatonic scale in solfeggio.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").