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.