Why would a drinking water dispenser on a sink have drainage lines?
I found a good explanation at:
http://www.purewateroccasional.net/hwairgapfaucet.html

The filter (possibly Reverse Osmosis) part has a drain pipe.
But it doesn't go directly to the drain.
Goes up to the tap base, where it dribbles in the open air (the "air gap") and back down through the other pipe to drain away.

If the second drain line becomes blocked, this "air gap" can leak.
Not sure if the possible leak would only occur when the tap is in use.