The words cosey, cosy, cozy sound the same but have different meanings and spellings. Why do cosey, cosy, cozy sound the same even though they are completely different words?
The answer is simple: cosey, cosy, cozy are homophones of the English language.
Obsolete spelling of cozy.
Variant of cozy.
Snug, comfortable, and warm.
Marked by friendly intimacy. See Synonyms at comfortable.
Informal Marked by close association for devious purposes: a cozy agreement with the competition.
To make oneself snug and comfortable: cozy up with the Sunday paper.
Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License, 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").