To my knowledge before my great grandfather Judah came to India, the custom in Baghdad was to refer to people as being the son of, or in Arabic 'ibn.' Thus, Judah would have been called Judah ibn Moshe. However, on Judah's arrival in India at the turn of the 20th century, it appeared that a surname was required. The need for a surname may have been due to the requirements of the British who ruled India at the time. He therefore decided to use his name as the surname and hence the Judah family was born.
I hope this helps
Best regards
Reuben Judah