Yes, merit money can cut into financial aid. It does depend upon the school and the types of aid as to how and how much.
Scholarships generally are used to reduce need, so when you get a merit/need package all from a college, they’ve usually done all of the reductions. Rare to get more than your need, but it can happen if the merit exceeds it or if the college does allow stacking. Some things one cannot stack. You cannot get subsidized loans, for instance, if your need is fulfilled by grants, even if some are scholarships.
Often schools have a standard aid package formula and when merit scholarships enter the picture, they are used to first reduce the subsidized loan portion, The work study part of the package. Though PELL grants are stackable, many colleges do not permit their own funds to exceed Cost of Attendance.
Where most people see how scholarships reduce aid is when they get outside or late awards, that then are set against the financial aid awards. Yes, it hurts when that happens. Financial aid at many schools is distributed purely for need so any windfall a student gets elsewhere can directly reduce it.