From what I see, the Cyrodiil Dark Brotherhood's myths shape the Sithis that they interact with, like with just about any non-Daedric god, so they get what they believe. In contrast, the Morag Tong are ultimately instruments of Mephala, a Daedra.
Yes, though what they got doesn't come fresh of the god-factory as a single never used before product shrink-wrapped and all. Like everybody else they end up some second-hand idea to fill in for it, it might be dressed in funny robes, black make-up and speak the cool tongue, it's still going to be tied that idea.
It's those ideas that spontaneously took shape in the first place because they -even without people- resembled some fundamental concepts like love, law or knowledge. Since it's those concepts that garner the power for action, you have to shape your myths to match them so you too can leverage their power.
Hence the statue of Sithis looks like Lorkhan. He's the guy that brought death back to the Auribs and together with Mephala, the mother of secrets, they gave birth to a cult dedicated to murdering people in secret. It's their combined concepts that make the Morag Tong and the Dark Brotherhood possible.