The system you showed may actually be the type that does not depend on the position on the pilot. Edit: The version you have works with a Moiré pattern, and is indeed dependent on which pilot is looking at it.
I dug into the AIS Netherlands information for our country's biggest airport Schiphol (EHAM/AMS). They have a few different systems; the one you link seems to be a lot like the SAFE DOCK system, which uses a Laser Range Finder (LRF) system to identify the position of the aircraft
Due to the digital display presentation, both pilots get the correct alignment information as well as the closing-rate and stop information.
I'm guessing that this system will become more common to allow for more flexible crew tasks during taxi. Furthermore, this system can detect obstacles, and as such provides extra safety.
The two other systems on EHAM do depend on the seating of the pilot. The effect of the wrong pilot parking the plane is very simple: your plane will be parked off-center with a deviation exactly equal to the spacing between the pilots: your plane will end up such that the co-pilot is sitting where the captain should be. The fact that the AIS does not quote any procedures for this says it all: you just shouldn't do it and wake the captain for a minute to do his duties.
Of course, there are procedures for when the system is not available (emphasis mine):
The system is operated by an employee of a handling company, who also keeps a safety watch during the docking. The pilot of an arriving aircraft has to be sure that the system is activated. If not, pilots should not enter the aircraft stand and stop before the red boundary line, until the system is activated or a marshaller has signalled clearance to proceed.
The logical conclusion we can draw here is that, if the systems are not available (or, in the case of the question, if the captain is fast asleep), a marshaller will guide the plane in - either way, a marshaller will stand by to oversee the docking and make sure the SAFE DOCK system is living up to its name.
As @reirab helpfully pointed out, the marshaller's job may be nothing more than a simple clearance to enter the gate and let the pilots do their job; they should be perfectly capable of parking an aircraft without the displays. In the aviation world, it's perfectly acceptable to spend a lot of money on these things if they can increase safety; if only due to reduced workload for the pilots.