Suggesting it or not, people filing for Saudi visas have claimed things that are not true in order to get a Saudi visa (including for purposes of the religious tourism visas) and most have apparently gotten away with it. To get caught would require really being an idiot or a glutton for punishment.

At least in some areas and at some times, Saudi visas have asked for religious identity even when the purpose of the intended visit had nothing to do with religious tourism visas (for the Hajj or otherwise). And those who indicated religion as Judaism when applying for various kinds of Saudi visas would often see the results of Saudi bigotry in action and be refused. Not always, but often.

Indian visa applications in recent years definitely ask for religion at least in India's largest OECD tourist source countries -- something that was not true for most of India's post-Independence history.