They use clay beads, which are an aluminium mesh with H and OH groups weakly bonded to it (a weak ionic bond like this is called a "Ligand bond"[1]).
Firstly, it filters out bacteria and "bits" because of the porosity of the mesh (works like pushing current through a complicated network).
However, additionally it filters out non-H-or-OH ions from the water, which have a stronger Ligand bond with the Al mesh. (Obviously you eventually run out of H and OH ions in the network, as they're all replaced, which is when you need to change your filter.)
As such, we used clay-filtered tap water as 'distilled' water in all our experiments, because the worst it contains is bacteria and dissolved salts. I can't speak for the Coca Cola though, that's weird. I'm sure organic molecules would be small enough to get between the beads but too heavy/not ionic enough to bond to the mesh, or something.
Because adsorption is a consequence of surface energy and weak bonding, I believe it's considered to be a chemical process, not a mechanical one.
Adsorption is due to weak bonds that form between molecules of a surface and a fluid. AFAIK this bond is not a 'new' chemical bond since it does not break / replace existing chemical bonds in both the adsorbed molecule and surface molecules.
Rather, in the case of carbon at least, the 'bonds' are the van der Waals forces.
http://www.bevreview.com/wp-content/image_kosherforpassoverc...
https://en.wikipedia.org/wiki/Caramel_color
That is, sugars that underwent Caramelization https://en.wikipedia.org/wiki/Caramelization