First of all, this type of time reversibility is the feature not of laminar flows per se, but of a special subclass of them: Stokes flows (also known as creeping flows). Creeping flow is described by linear system of equations (Stokes equations) that have the property of instantaneity, that is the solution depends on time only through time-dependant boundary conditions.
This means that once (as in video) you stop moving the handle the movement of fluid instantly stops. Coupled with linearity of equations this ultimately means that the current position of fluid particles (including colored fluid) in the jar depends only on the angle of rotation of the handle and not on how it was achieved, that is not on speed of handle movement. This means that if the direction of handle rotation will be reversed the flow will also be reversed and once the handle resumes its original position the colored particles also should be in their original place.
However, this description in terms of creeping flow is only approximate as demonstrated in video since the form of colored blobs is deformed in the end.
As for thermodynamics, while it may seem that we have reversal of fluid mixing, in this case no actual fluid mixing occurs because the colored fluids in the 'mixed' state are actually aligned in spiral-like shapes along the walls of the jar and differently coloured spirals though close, do not intersect and so no mixing occurs.
And, of course, we do not need to involve quantum mechanics in explanation.