How many particles come in? Well, if the plate has area A and the average velocity of these particles is u, then the volume of incoming particles interacting with the plate in a given time t is just u A t. This means that the mass interacting per unit time is just ρ u A. Thus the force needed is:
F = ½ ρ u A v λ/d
Thus μ = ½ ρ u λ, or just η = ½ u λ.
In other words, the diffusivity of momentum η doesn't care about how fast the plates are going, but is instead simply proportional to the size of the boundary layers, which is the mean free path, and the rate at which particles move, which is the average atomic velocity.
Thickness is a sort of subjective expression of how easily momentum is diffused. Something is thick if, when you stick your hand in it and move it around, it is hard to move your hand around.
Usually we just say that the thickness is the viscosity, but I suppose a better treatment involves treating both forces. If you push a fluid around, you push a mass ~ ρ v A at a speed v, leading to an "inertial force" ρ v² A. There is also a viscous force, μ A v / d. Thus our perception of thickness is based on:
Av (ρ v + μ / d).
Assuming that A and v are just properties of our arm or finger and don't figure into our final judgment of what is thick or thin, the characteristic quantity is just ρ (d v + η). On a human scale, we might swirl a stick 1cm wide at about 1 m/s or so, so that v d ~= 10^(-2) m²/s. On the other hand, for most fluids, η is closer to 10^(-6) m²/s. (mercury is 10^-7, water is 10^-6, air is 10^-5, oil is 10^-4, glycerin and honey are 10^-3, sour cream is 10^-2 or so.)
This means that most of the time, when we're making the judgment, we're making it pretty much based on ρ, which is also of course a part of viscosity (μ = ρ η) but we are actually perceiving the inertial force much more than the viscous force. It's only when you're, say, mixing guacamole that your perception of "thickness" begins to be a perception of viscosity proper and not simply density. If you stirred mercury you would probably find it thick even though the Reynolds numbers would be lower by comparison.