It probably was not under a cabinet when it was installed. Floor drains were originally installed to catch water overflow in order to protect the downstairs from damage. They are problematic because the water in the trap evaporates eventually allowing sewer gas to enter the room. I would not eliminate it until it is looked at by a carpenter, but if it is under your new vanity, it may have to be moved. Fill it with mineral oil if you keep it to block sewer gas

Thanks for the input, Jime :-) However, what makes this somewhat strange is that our house is a one story ranch, so there is no downstairs to protect. The cabinetry was also original to the house - built in place by the same person who did the kitchen cabinets, and who didn't do a particularly great job anywhere. We have already replaced the vanity in the master bathroom and that room doesn't have this issue.

We are certainly going to get a plumber to look at it before we do anything - I just couldn't think of a logical reason for this particular set-up. At least in the laundry room, the drain is set into a sunken area which contains the water and directs it towards the drain. In the bathroom, the water flows away from the drain!

I wouldn't mind keeping the drain, if we can resolve the back-flow issues, because we will be installing wall-mounted sinks and can tile around it.