My understanding was that they used a false name to leave Kenya for the West. As former members of the Somali regime they couldn’t get visas, so they took the name of a family that did have permission.
That seems to have dropped through the memory hole.
Like others have said, lying, cheating and stealing are endemic to the Somali Culture, Ilhan Omar is likely not her true name, this is just one of many lies that exist around her.