Wow that is creepy. You are right, it sounds like an animal crying faintly in the distance. But if it were, it wouldn't go on for 18 months without changing. If you had animals living under your floorboards you would at least hear seasonal cycles. Although when I was a new parent, I thought that our baby would never stop crying, at least she didn't cry for 18 months straight. Wow.

So it must be mechanical, plumbing or electrical.

It can't be anything battery powered... unless... do you have a battery powered clock in that bathroom? I know I've had clocks which made faint unusual sounds sometimes.

Failing that, it has to be either something on the mains power, or it's something that just has a natural cycle, like some kind of thermal cycling. I would have suspected something related to the air flow in the house due to the furnace, but you said it happened with the furnace off. It can't be the house creaking, that wouldn't happen 24/7/365, it would go in cycles with the temperature, the wind, and the seasons.

As Mark said, cut the main power to the house and see if it still occurs, to narrow down whether it's caused by something that's plugged-in. Also listen more closely, move around the room, listen to hear if it's coming out of a drain pipe or an air vent, try to narrow it down further.

Is there some appliance, such as a water heater, in or near that bathroom? Could be the pressure release valve on the water heater, venting pressure and then letting it build again, at regular intervals. Or any other place where there might be that kind of pressure/build/release cycle. I know that some houses have a pressure regulator valve for the water at the point where the water pipe enters the house, maybe it's that.

If you're on natural gas supplied by the city, perhaps it's related to the pressure regulator for the gas, maybe that's just outside your bathroom. Also look and see if the electric or gas meters are on an outside wall near that bathroom, maybe one of them is making the noise.

Really interesting problem.