The words fay, fey sound the same but have different meanings and spellings. Why do fay, fey sound the same even though they are completely different words?
The answer is simple: fay, fey are homophones of the English language.
To join or fit closely or tightly.
A fairy or an elf.
Archaic Faith: "Sirrah, by my fay, it waxes late” ( Shakespeare).
Having or displaying an otherworldly, magical, or fairylike aspect or quality: "She's got that fey look as though she's had breakfast with a leprechaun” ( Dorothy Burnham).
Having visionary power; clairvoyant.
Appearing touched or crazy, as if under a spell.
Scots Fated to die soon.
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.