Both! They could be rivals... This could tie into the whole Imperialist/Separatist dynamic. Perhaps for some reason, the Dark brotherhood would prefer that the Empire remain in control of Skyrim, while Morag Tong doesn't.
Why would the DB want a stabilizing force? If you have read anything, they enjoyed the greatest amount of business when there was no imperial government between the fall of the 2nd empire and the rise of Tiber Septim. The Dark Brotherhood would see much more business if everyone was separate and not unified under a single banner. Plus, they'll kill just about anyone and work for just about anyone, for a price.
Also, the Morag Tong would be heavily weakened due to the Red Year and argonian invasion. In addition, it's very likely they were hunted down upon entering Skyrim by not only the Dark Brotherhood, but the government(s) of Skyrim. They no longer have Vivec giving them special protection of legally murdering members of Great Houses, with occasional BS writs (aka gray writs). Not to mention the MT lived in pretty open accommodations, never having to live as an underground organization. They were nothing but killers who lived out in the open, while the DB has been successfully living underground since the fall of the Reman empire, and have more experience honing their skills in the silent arts of killing.
At this point, the Morag Tong are completely illegal everywhere, and with the dunmer becoming nothing but refugees, holding onto an assassins guild would likely cause more harm than good, especially if members of the Morag Tong were caught. It would give a good reason for city leaders to remove the dunmer refugees. Also, inter-warfare would do nothing but weaken the dunmer more and more.