I very much doubt that people will even want to give out information about themselves that they actually play gambling games in online casinos. In society, as a rule, there is a bias towards people who play in casinos. Therefore, in some cases, even the fact that someone will know about your online casino games will be enough for a person to stop playing in an online casino.
Government or any public or private organization should not intervene or interrupt in any activity that people are doing that may have a bad impression in the public, the general society has a bad impression on gamblers and this will discriminate them, they can just pose serious warnings but it's on the person to imposed it to himself not the government.
Its government right to regulate anything inside their land, even in democracy you can't ask questions if it has been made a law.And not much people going to consider themselves as gambler community,they will just move to online gambling where there is no restriction and no documentation.