Going with my gut this is my reasoning for the faction split:
The Runners are based on the ICE breakers: Shapers are particularly good with Code Gates, Anarchs with Barriers, Criminals with Sentries. This split also creates the factions themselves. Code Gates in tend to be cheap for their effects but don't tag or damage the runner (Tollbooth, Enigma) so Shapers end up being the "Consistent Cash Flow" faction. Barriers end the run and act like barriers, so Anarchs are built to break the barriers down; they're the "Trash" faction. Sentries are built around nasty effects, traces, and tags, so Criminals are anti-tagging and burst Cash Flow to pay off the Traces.
The Corps are based on the 4 major effects you can put on the runner: Net Damage for Jinteki, Meat Damage for Weyland, Brain Damage for Haas Bioroid, Tagging for NBN. Splitting the factions like this colors what different factions could do for balance reasons. Brain Damage is easily the best type of damage, so Haas Bioroid's ICE has a built-in back door; HB also is the only faction that can't reliably win by flat-lining the runner. Net Damage is usually about slowing the runner down rather than winning, which is why Jinteki ended up with Nisei MK II. Tagging is about slowing the runner down in a different way: while Jinteki forces the runner to spend clicks drawing cards, NBN makes the runner spend clicks taking credits; NBN ends up being the Speed faction with cards like SanSan City Grid that the Runner can't really afford to Trash if the Corp is doing his job right. Meat Damage is about flatlining, so Weyland is built around the only card in the game that can consistently flat-line: Scorched Earth; to balance Weyland, the Corp must go out of faction to get good tagging tech or must sacrifice agenda points for his best ICE or effects.
That's my 2 cents for why the factions are split this way.