It's pronounced the same only by people speaking English with a French accent. An American, Brit, Indian, or any other native speaker of English absolutely does not pronounce "bit" the same way a French person pronounces "une bite."
Rather than measuring whose French pedigree is longer, I will put down a wager on this. ₹3? :D
I don’t doubt that you speak French. French people tend to have difficulty distinguishing those sounds because they are not distinguished in French. In English, they are: English has a much larger inventory of distinct vowel sounds than French (or indeed most European languages). In typical French-accented English, “bit” is indeed pronounced like the French word “bite”, but in native speaker English, it is not.
I am a native speaker of American English and also speak French quite well. If you neither accept personal experience, nor what is written on Wiktionary, what evidence would you accept?
Rather than measuring whose French pedigree is longer, I will put down a wager on this. ₹3? :D