Yes, some schools do have merit within need, but I’d be surprised if Stanford and Williams do. Neither give merit money. It’s rare that an institutional formula would be that much higher than the FAFSA EFC. A school is not permitted to give a dime over the FAFSA EFC if it also wants to give out federal money including work study and subsidized loans. By giving out more than the EFC, it loses the option to give the student any federal funds and has to full y fund that student themselves. Something doesn’t add up here.
Giving money to visit is not unusual. Schools often offer this, even to a parent to accompany when looking for a student for geographical diversity purposes. My son and DH were offered the cost of plane tickets to come visit a school when we did not even qualify for financial aid.