They do this because they are aware of all the information leaking incidents which happen on a regular basis which is why they prefer gambling anonymously.
You might be right and it might be right for few cases but in maximum cases this cant be true. And mainly the purpose of using fake document in gambling site is to misuse bonus system. Another reason might be to hide him/her from being expose from legal action. And only few percentage people do not use it for the fear of leaking incidents.