Must be state-issued ID (Driver's License or non-Driver's License card from state DMV, passport, or military ID with DoB on it only. Also the ID card MUST NOT be expired, especially on cash trades.) Anyone looking under 30 is supposed to be carded.
Some managers may make it mandatory for all ID's to be seen regardless of age, this is usually during big M-rated game weeks like Battlefield, Halo, or Call of Duty but that is YMMV.
If an employee is caught selling an M-rated game to a minor (ESRB sting, secret shopper, etc.) clerk is auto-fired and store manager gets final warning write up.
Also you must be 18 to do cash trades (depends on state law but that's fairly standard) and you must surrender your ID for the employee to take down all info (specifically the ID number) so that's probably why the mother's ID was taken for the kids trade-in the above poster talked about.