The words shoe, shoo sound the same but have different meanings and spellings. Why do shoe, shoo sound the same even though they are completely different words?
The answer is simple: shoe, shoo are homophones of the English language.
A durable covering for the human foot, made of leather or similar material with a rigid sole and heel, usually extending no higher than the ankle.
A horseshoe.
A part or device that is located at the base of something or that functions as a protective covering, as:
A strip of metal fitted onto the bottom of a sled runner.
Used to frighten away animals or birds.
To drive or frighten away by or as if by crying "shoo.”
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").