Last year I resigned as a Front Desk Manager at a Marriott hotel in the Northeast. We would have this request quite often.
We'd generally require the reservation to be made in the exact name of the person checking in (unless there were notes stating otherwise), but at checkin it was not uncommon for a request to be made to have it so no one could locate them there.
We'd change their name to Duck/Daffy or something similar and let them know that would be appearing on their bill, while at the same time maintaining their real name on an internal field in our system on visible to us.