Liquids confined to molecular scales become anisotropic and often show pronounced self-organization such as layering. Although this effect is well accepted, it is still debated if confinement induces measurable changes of viscous friction. We use molecular dynamics to address this issue by simulating a Lennard-Jones liquid confined between a solid cylinder and an atomically smooth surface. The simulations reproduce the well-established variations of normal force, density, and diffusivity with the distance between wall and cylinder. We find high diffusivity and low density when the numbers of layers is in between integers. This observation seems to contradict most experimental results on the effective damping between atomic force microscope tips and substrates when interpreting them within continuum hydrodynamics used to connect liquid viscosity and diffusivity. This contradiction is resolved by directly extracting the damping that the tip experiences, which we achieve by using the fluctuation-dissipation theorem; as in experiment, we find local minima in the damping near integer numbers of molecular layers and maxima in between. These variations correlate with distinct structural changes in the microscopic order of the fluid. We reconfirm that constitutive equations valid at macroscopic scales cannot be used to interpret confined liquids and finally conclude that viscous friction displays measurable, non-monotonic behavior with the degree of confinement.