The words fane, feign, fain sound the same but have different meanings and spellings. Why do fane, feign, fain sound the same even though they are completely different words?
The answer is simple: fane, feign, fain are homophones of the English language.
Happily; gladly: "I would fain improve every opportunity to wonder and worship, as a sunflower welcomes the light” ( Henry David Thoreau).
Archaic Preferably; rather.
Archaic Ready; willing.
Archaic Pleased; happy.
A temple or sacred place.
A weathercock, a weather vane.
To give a false appearance of: feign sleep.
To represent falsely; pretend to: feign authorship of a novel.
To imitate so as to deceive: feign another's voice.
To fabricate: feigned an excuse.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition, from Wiktionary, Creative Commons Attribution/Share-Alike License and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.