The words coo, coup sound the same but have different meanings and spellings. Why do coo, coup sound the same even though they are completely different words?
The answer is simple: coo, coup are homophones of the English language.
To utter the murmuring sound of a dove or pigeon or a sound resembling it.
To talk fondly or amorously in murmurs: The visitors cooed over the newborn baby.
To express or utter with soft murmuring sounds.
A brilliantly executed stratagem; a triumph.
A coup d'état.
A sudden appropriation of leadership or power; a takeover: a boardroom coup.
Among certain Native American peoples, a feat of bravery performed in battle, especially the touching of an enemy's body without causing injury.
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").