The words oh, owe sound the same but have different meanings and spellings. Why do oh, owe sound the same even though they are completely different words?
The answer is simple: oh, owe are homophones of the English language.
Used to express strong emotion, such as surprise, fear, anger, or pain.
Used in direct address: Oh, sir! You forgot your keys.
Used to indicate understanding or acknowledgment of a statement.
To be indebted to the amount of: He owes me five dollars.
To have a moral obligation to render or offer: I owe them an apology.
To be in debt to: We owe the plumber for services rendered.
To be indebted or obliged for: owed their riches to oil; owes her good health to diet and exercise.
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.