It would be just a matter of time. EY and EK will have similar strategies of spreading their individual names with minor differences.
I major difference is certainly the name Emirates and the UAE flag on the tail. It shows to the general air travelling public that EK is the real flag carrier. Etihad would probably like to swap liveries and names for the time being. Eventually, they will "unite" and make it Etihad-Emirati with 2 major hubs and spokes.
Dubai is also much more cosmopolitan than Abu Dhabi, therefore EK flies to more Western destination and remember they are almost 20 years older than EY.