The story is light on details so I ass/u/me some things. The copyright infringement was likely due to torrents, i.e. from the internet's point of view, addresses owned by Cox were publishing/hosting content (under the hood: really Cox's customers seeding torrents).
So if I were an MPAA/RIAA -member company, I'd send Cox a DMCA notice ("Cox, stop sharing my copyrighted work") which really means "Cut that customer off or otherwise make them stop, or else get a DMCA counternotice from them, so I can go after them instead of you." And if that's what happened, then it sounds like Cox said no (didn't make it stop and also didn't pass the buck to their customers. So they sued Cox instead of Cox's customers.
But that's based on assumptions and speculation, hence my question. But yes, I know what a DMCA notice is and I think that mechanism was likely in involved at some point in the story.