Why does activated carbon preferentially adsorb anions?
This answer applies to carbon filtration of water.

According to most of the sources I found, activated carbon binds to most substances through London dispersion forces (from the Wikipedia article). This should mean that it adsorbs larger molecules and non-polar molecules preferentially since they would have larger dispersion forces. As your quote indicates, it does not bind polar molecules like "alcohols, glycols, strong acids and bases" well. This source also lists neutral, non-polar compounds, along with organics and compounds with low water solubility as being effectively removed. 

I think that you may have misread or misinterpreted the sentence about binding "chlorine and iodine" as saying that it traps chloride, iodide, and (by extension) other anions well. I can't find data to support this statement. It traps molecular iodine well, and as a large (in terms of electrons), non-polar molecule, that makes sense. But it removes molecular chlorine $\ce{Cl2}$ through a chemical reaction by reducing it to the chloride ion $\ce{Cl-}$ which is then soluble in water and flows through the filter.

I did find a couple of sources (1, 2) that said "activated charcoal" has a slight electropositive charge that helps it to attract negatively charged species, which would fit with your statement that it preferentially attracts anions. (However, they don't list what kind of negative species they're talking about and nitrate, sulfate, and fluoride ions were specifically excluded). The idea that it will remove ionic compounds seems to be contradicted by the statement in at least a couple of sources (one given here) that carbon filters don't remove minerals, salts, and dissolved inorganic compounds