they verify things that you claimed points for, simple. - YES this is the eligibility checkCorrect, they verify things that you claimed points for, simple.
and there is BACKGROUND CHECK where everything about the applicant will be verified, there is no limits, since IRCC isn't disclosing their complete background checks scrutiny and procedures we don't know to what extent they'll go for this check. IRCC just publish the percentage of rejected cases but will never publish the reasons. When it comes to UK the background check is so strict that it checks the applicant's online and real life activities, since their priority of safeguarding populated cities they will even go the point of gathering IP addresses, work places, and financial activities includes history of money transfers, wire transfers etc. And I believe the USA and Canada will look into such backgrounds when it comes to determining an individual's admissibility to their country.