I thought I was the only one whose eyebrows went up when a show about men and women who are strangers moving in together was "ABC Family" programming.

Hard to believe this started out as Pat Robertson's network before it was sold several times.
Crusder, you are on the right track, sort of. The cable company from which you buy your service and the cable company (distribution) that actually provides the programming (the "station") are sometimes one and the same. And, the four broadcast networks ABC, CBS, NBC and FOX all also have cable channels and/or also own cable distribution systems. The local cable companies are also dwindling down into just a few mammoth companies....who are also now becoming the providers of cable internet access as well. Some of the cable companies are also looking into buying the satellite TV companies, too.
Soon there will be just one big CABLE COMPANY controlling it all.
There are two types of cable networks. Basic cable networks sell advertising (ESPN, TLC, etc., etc.) Premium networks (HBO, Showtime, etc.) do not sell advertising but charge a premium from the subscribers (that is collected by the local cable company who also takes a cut). The basic cable networks are also paid money from the cable systems that carry them because they allow the cable companies to also sell advertising and "cut-in" to their programming (like a local TV affiliate). Some local cable companies look to recoup this cost from the subscribers with their pricing.
Back in the 1970's when cable was just catching on, CBS had to divest itself of its cable interests. The company that was formed was Viacom. Fast forward to the late 1990's when cable regulation is a thing of the past and a lot of twists and turns have CBS and Viacom as one company again.
***Updated 7/30 - I corrected the premium cable network to HBO -- must have had ABC on the mind. Sorry.****