The noted writer Marv Wolfman notably writes about this…
Mark, I'm not a lawyer nor do I play one in the comic books, but this doesn't sound truthful. Both CBS and Comedy Central are owned by Viacom. So are they saying if Colbert doesn't stop playing Colbert, Viacom will sue itself? Huh?
Well, different divisions of the same company do sometimes sue each other — or at least threaten to — and they often fight over which of them controls and/or is credited with the revenue derived from certain assets. You, Marv, have witnessed the comic book division of Time-Warner (DC Comics) fighting with the animation or movie divisions of that same conglomerate over who gets to make certain decisions about Superman, Batman and other properties and which division will receive what portion of the revenues from a given Superman or Batman TV or movie project. If you head up a division, it's in your interest to see that your division looks profitable on paper. You don't just think, "That doesn't matter. It all goes to the same corporation."
However, in this case, CBS and Viacom are (somewhat) two separate companies. They weren't but now they are because in 2006, Sumner Redstone — who owns like 80% of them together or apart — decided they should be apart. In the 20% of CBS that Redstone and his family don't own, there are stockholders who own none of Viacom and in the 20% of Viacom that the Redstones don't own, there are stockholders who own 0% of CBS. Ergo, they are run as two separate companies and it is vital to clarify which assets go with which company. Everyone expects at some point, Redstone will decide it's in his best interests to have them be one again but he could sell one or the other. (See this article for more on that.)
Also, I would imagine the cease/desist warning to Colbert was a matter of the corporation wanting to remind him that they own something in the way of intellectual property relating to The Colbert Report. Say he decided some day to make a movie as "that" character. They want him to be on notice they have a claim on it and would expect financial involvement at least and maybe some form of ownership. It gets murky because they are laying claim to his real name and likeness in that role (the way Paramount owns the likeness of Mr. Spock, which happens to involve Leonard Nimoy's face) and to an attitude Colbert could well invest in a new character not named Stephen Colbert. So it's kind of a bargaining position. It's easier to claim you own 100% of something and then to back that claim down to a lesser percentage than it is to do it from the other end.
People sometimes think this matter of the network asserting it owns intellectual property is piggish but I.P. does have a value, often a considerable one. NBC has made a lot of loot from movies like Wayne's World and The Coneheads because they retained rights to intellectual property created on their dime for Saturday Night Live. If I were Viacom, I wouldn't be so quick to give up all claims to The Colbert Report. At the very least, I'd think that if Stephen Colbert wanted to use that character again and say it's the exact same guy, we could get something valuable from him — a contractual concession, some guest appearances, something — in exchange.