Mos, you are correct. The controller made this mistake, after the EK flight made their mistake in misreading the clearance.
He then gave extensive and precise traffic information to both aircraft. At that moment, the controller effectively corrected his mistake. EK acknowledged, but did not pick up the conflict, their mistake again.
Concerning being on the same frequency, if you listen to the recording attentively, it would seem that the indicators (voice, timeline) point very much to all three participants (controller, EK ,SEY) being on the same frequency, rather than the controller eventually speaking on two frequencies to two different aircraft crossing on the same airway.
But in all fairness, i cannot be sure, so you are correct again.
(arketip:typo! mine!)